Mistakes, Errors, Other Minds, II

There are mistakes and errors that it is, from an educational perspective, wrong to dismiss as such, wrong to regard merely as matters for correction. “Sean numbers” constitute one example—a particularly striking one from an intellectual, and even a moral, point of view, in that Sean’s observation is a novel one for others, not only for him. He sees number freshly—perhaps we can too. He makes a discovery or, indeed, according to Ball’s reflection at the time, an “invention.” He adds value to the common store.

Though Sean’s move is unusual, this general phenomenon—the mistake or error containing an accurate, sometimes profound, observation or insight—is a familiar one to teachers of, for example, literacy. Errors and Expectations, Mina Shaughnessy’s influential account of the work of and pedagogy appropriate for “basic writers” (students arriving in college essentially unschooled in writing), analyzed with great power the logic and, as she insisted and demonstrated, the “intelligence” of students’ mistakes. Likewise, scholars of early literacy coined the term “miscue” to underscore the sense and logic of a vast number of beginners’ mistakes, and to avoid the pejorative connotations of that designation. (An example of a miscue is the beginning reader’s substitution of “house” for “home,” when the latter is what actually appears in the text. In a case like this, the teacher is much better able to serve the learner when he recognizes that in reading “house,” the beginner has made semantic sense, has gotten the initial phoneme right, and is syllabically and rhythmically accurate as well.)

A form of the general phenomenon is also recognizable in the domain of what’s generally called “classroom management.” Although Winnicott’s interpretation was not made in a classroom context, instances of stealing and analogs of it occur often enough in classrooms, and readings akin to his reading of theft are often productive. Readings, for example, that take the form of asking of a child’s misbehavior in school, “what need might this action be expressing?”—or, to continue the theme of mistakes and errors, “mis– or mal– expressing?” Children who habitually careen into others, grab onto them, appropriate their coats or hats, and so on, may sometimes be usefully read as seeking human connection in the school (and, following Winnicott, believing they might find it, while yet being unskilled or ambivalent about finding it). It can make sense to understand children who “tattle” as expressing a need for clarity about the moral order, or the power structure, or both, of the classroom. And so forth. Any teacher with a little experience will be able to supply dozens of examples. (At this point, I suppose I am obliged to stipulate that to interpret children’s actions in these ways—which in any case can only be begun, but not concluded, in the absence of specific and contextual information—does not require one to condone or excuse the particular behaviors.)

In all these sorts of occurrences, these human events that are comfortably enough, but not so helpfully, called mistakes, errors, errors in judgment, mis-steps, and so forth, it is possible to see meaning not nonsense and, more than that, logic, a kind of forward movement, an application of the mind or will to a puzzle or problem.

This returns me to Green’s “Building a Better Teacher” article, and to its central concern, whether or not good teaching can be learned. Green writes of one of her two key subjects—teacher educators who differ in numerous interesting and significant ways—that “all (of Doug) Lemov’s techniques depend on his close reading of the students’ point of view, which he is constantly imagining.” In one of several very satisfying symmetries in the article, the second of Green’s central characters, Deborah Ball—once Sean’s teacher, now Dean of the University of Michigan School of Education—is quoted, several paragraphs later, as saying “Teaching depends on what other people think, not (on) what you think.” That’s a lovely thought and a true statement. It, and the fine characterization of Doug Lemov endeavoring to imagine the students’ points of view, directs us squarely to one of the major tasks of learning to teach: Learning to perceive what others think, learning to imagine what and how others see, learning how to make fruitful contact with other minds. Mistakes, errors, and their kin—taken up with interest, curiosity, a serving or two of patience; taken up as likely manifestations of the mind in search mode—can be powerful material for such learning.

Mistakes, Errors, Other Minds, I

Stealing can be “a sign of hope,” child-psychoanalyst D. W. Winnicott once wrote. He meant that a child’s petty theft can be read as having little or nothing to do with the thing taken, and everything to do with the acts of reaching out and taking hold. The theft can be read, that is, as expressing the child’s hope that some way of addressing his needs does in fact exist and may eventually be found. The child in this situation, Winnicott writes, “is looking for the capacity to find, not an object.” The passive or depressed child, by contrast, may be understood to communicate that she has lost hope that she can find what is needful, lost hope, perhaps, that the world, or at least, that part of it constituting her immediate context, is in any way capable of answering to her vital needs.

I was reminded of Winnicott’s idea when reading the very interesting article, “Building a Better Teacher,” by Elizabeth Green, in the New York Times Magazine of March 7. There is much to like in this article, and much that’s worth discussing, but I was prompted to think of Winnicott by one particular passage, and that is what I want to focus on here. In this passage, Green describes an episode (well-known to those who study mathematics education) in an elementary school math class. The connection I find between her account of this episode and Winnicott’s analysis of childish stealing is two-fold. On the one hand, there is the appearance of mistake or error; on the other, there is the problem of interpretation. In the case of Winnicott, the analyst encounters a familiar sort of behavior, but does not take its meaning to be self-evident; rather, he treats it as a call to interpretation. (Notably, the path to interpretation in practice such as his is in great measure a course of unhurried listening.) Reverting to my first hand, if Winnicott’s interpretation is correct, the stealing child may be said to have made a mistake (he has taken the wrong thing: that bit of electronics is not going to satisfy) or to have made an error (he has strayed: this path is a dead end, other paths are better).

The episode Green describes can also be understood as an instance of a child making a mistake; certainly, she appears to see it that way. (It is also a rather special case of pedagogical interpretation, initially expressed as pedagogical tact, then as mathematical confidence, but I want right now to focus on the child.)

As Green tells us, in a third grade math class that has recently been studying the properties odd and even, Sean announces, “I was just thinking about six. I’m just thinking, it can be an odd number, too. ‘Cause there could be two, four, six, and two – three two’s, that’d make six! And two three’s. It (six) could be an odd and an even number. Both!” Green notes that Sean’s teacher, Deborah Ball, does “not…contradict him.” “And,” she says, “he (goes) on not making sense.”

But (the many virtues of the article notwithstanding), the statement that Sean isn’t making sense does not do justice to him, nor to this scene, which might better be understood as a perfect drama of sense-making. This is sense-making in which Sean’s mistake—his application of “odd” and “even” as if they constituted a more flexible pair of categories than is the case—is in fact a marker of his intelligence at work. The leading character in this drama of sense-making is of course Sean, but his class-mates and, critically, his teacher, Ball, play significant parts.

As Green reports, rather than contradicting, brushing off, or ignoring Sean, Ball, enacting a well-established custom of this intellectual community, invites the rest of the class to comment on his idea. Over the remainder of the discussion, everyone, Sean included, comes to agree that “odd” and “even” are mutually exclusive: A number is one or the other, not both. But the class also follows Sean in noticing that there are even numbers that are themselves the product of an odd number and two (e.g., 6 = 3x2, 10 = 5x2, 14 = 7x2), and that the occurrence of these numbers is predictable. Ball eventually proposes that numbers with this property be named “Sean numbers,” a suggestion that is well-accepted.

Sean had really been making plenty of mathematical sense. He had been learning about one observed regularity in the domain of number, the properties we call “odd” and “even,” and the distribution of numbers exhibiting these properties. With this as context—more precisely, with these properties constituting a new, somewhat untested intellectual tool—he had come to discern another regularity. Quite sensibly, given the genesis of his observation (and given that all observations have antecedents), he at first conceived of “his” regularity in the same terms, “odd and even,” as the canonical one he had recently been studying. Some thing or things—logic? elegance? social pressure?—persuaded him to agree to consider the recurring feature he had observed as a distinct numerical phenomenon, not as a further definition or potentiality of the “odd/even” pattern. All well and good. But at the heart of the scene is Sean’s discernment of a verifiable regularity in the world of number. Once pointed out, it was discernable by and comprehensible to others. Indeed, as Ball subsequently learned and shares in one of her many articles, at least one ancient mathematician had previously noticed and drawn attention to this same regularity. That “Sean numbers” were novel to his classmates, teacher, and most of Ball’s students and readers, and that they have not, neither before nor since, proven to be of enduring interest or usefulness to the community of mathematicians, does not amount to him “not making sense.” On the contrary, he made excellent sense. He was, at once, mistaken (in his application of a category), and (in his perception of a form of order in the world of number) on the mark.

Such a combination of mistake or error—often consisting of a failure to fully grasp a convention—and accuracy, sometimes profundity, of observation or insight, seems to me both a common and a precious symptom of a mind in high gear.

To be continued.

Education for what, II

I agree so whole-heartedly with Anna’s thought that “social aspects are integral and necessary to the primary purpose of education” (not to mention appreciating her kind comment) that I am tempted to say “yes” and leave it at that.

But that would be a kind of cheating. So, of the dozens of observations and claims that might be appropriate here, I will narrow the field and stick closely to the question of the social dimension of educational purpose, from two angles. I'll try to be brief.

In the first place, we are inherently social creatures, and indeed much of the pleasure we can learn to take in uses of mind is to be found in our interactions with other persons, other minds, both those in the live give and take of the classroom and those we meet in books and paintings and buildings and songs and human products of all sorts. Furthermore—and also, I should like to say, right next door, no, in the very same room—the powers of mind I spoke of assuredly, centrally, include the power of mind to manifest itself in work that is moving to others, the power of mind to frame an argument that persuades another to think or act differently, and so forth. “Power,” in other words, is a quality of human interaction with the world; learning to use the mind with power, then, must include learning to use it in ways that enter into the consciousness and conduct of others.

The second line I want to take today on the social dimension of educational purpose corresponds to the last point, or both of the last points, then—and that is to say that a purpose of formal education is always, somehow, to fit the learner for the world in which she finds herself. I mean this descriptively, and believe it applies to the most odious as well as to the most ennobling, and the most banal, programs of formal education ever conceived—although, generalizations being error-prone as they are, I’m very likely over-looking something in this claim, possibly something important. I look forward to learning this. In any case, the power of the learner—that is, the potential for consequentiality in the learner’s thoughts and actions—is therefore always at issue in formal education. Normatively speaking, the social purpose of education that I subscribe to, specifically, a purpose of public education, is to help learners assume the rights and responsibilities of democracy. That’s a bland formulation, I accept, one that begs more than it discloses, though also useful I think. To liven it up, I’ll subject it to some interpretation. To start, I’d say that “assume” implies “understand,” “take on,” “practice,” and if possible, “welcome.” These I hope begin to give some definition and difficulty (if we take these as tasks for teaching) to the seemingly bland statement. I add: The foundational meaning of democracy is, the people rule. To affirm democracy, then—to assume it, civically, and pedagogically, let’s say—must mean, to claim that “the people,” all of them, are, at base, wise enough, imaginative enough, compassionate enough, just enough to be fit to rule. From this it seems to me a short step to say that a purpose of education in a democracy is to help each child begin to realize in himself, and begin to recognize in his peers, those qualities of wisdom, imagination, compassion, and justice that underwrite the democratic project.