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.