He looked up at Ball, who was sitting in a chair among the students, wearing a black-and-red jumper and oversize eyeglasses. She continued not to contradict him, and he went on not making sense. Then Ball looked to the class. “Other people’s comments?” she asked calmly.

At this point, the class came to a pause. I was watching the video at the University of Michigan’s school of education, where Ball, who has traded in her grandma glasses for black cat’s-eye frames, is now the dean — and one of the country’s foremost experts on effective teaching. (She is also on the board of the Spencer Foundation, which administers my fellowship.) Her goal in filming her class was to capture and then study, categorize and describe the work of teaching — the knowledge and skills involved in getting a class of 8-year-olds to understand a year’s worth of math. Her somewhat surprising conclusion: Teaching, even teaching third-grade math, is extraordinarily specialized, requiring both intricate skills and complex knowledge about math.

Image Credit... Illustration by R. Kikuo Johnson

The Sean video is a case in point. Ball had a goal for that day’s lesson, and it was not to investigate the special properties of the number six. Yet by entertaining Sean’s odd idea, Ball was able to teach the class far more than if she had stuck to her lesson plan. By the end of the day, a girl from Nigeria had led the class in deriving precise definitions of even and odd; everyone — even Sean — had agreed that a number could not be both odd and even; and the class had coined a new, special type of number, one that happens to be the product of an odd number and two. They called them Sean numbers. Other memorable moments from the year include a day when they derived the concept of infinity (“You would die before you counted all the numbers!” one girl said) and another when an 8-year-old girl proved that an odd number plus an odd number will always equal an even number.

Dropping a lesson plan and fruitfully improvising requires a certain kind of knowledge — knowledge that Ball, a college French major, did not always have. In fact, she told me that math was the subject she felt least confident teaching at the beginning of her career. Frustrated, she decided to sign up for math classes at a local community college and then at Michigan State. She worked her way from calculus to number theory. “Pretty much right away,” she told me, “I saw that studying math was helping.” Suddenly, she could explain why one isn’t a prime number and why you can’t divide by zero. Most important, she finally understood math’s secret language: the kinds of questions it involves and the way ideas become proofs. But still, the effect on her teaching was fairly random. Much of the math she never used at all, while other parts of teaching still challenged her.

Working with Hyman Bass, a mathematician at the University of Michigan, Ball began to theorize that while teaching math obviously required subject knowledge, the knowledge seemed to be something distinct from what she had learned in math class. It’s one thing to know that 307 minus 168 equals 139; it is another thing to be able understand why a third grader might think that 261 is the right answer. Mathematicians need to understand a problem only for themselves; math teachers need both to know the math and to know how 30 different minds might understand (or misunderstand) it. Then they need to take each mind from not getting it to mastery. And they need to do this in 45 minutes or less. This was neither pure content knowledge nor what educators call pedagogical knowledge, a set of facts independent of subject matter, like Lemov’s techniques. It was a different animal altogether. Ball named it Mathematical Knowledge for Teaching, or M.K.T. She theorized that it included everything from the “common” math understood by most adults to math that only teachers need to know, like which visual tools to use to represent fractions (sticks? blocks? a picture of a pizza?) or a sense of the everyday errors students tend to make when they start learning about negative numbers. At the heart of M.K.T., she thought, was an ability to step outside of your own head. “Teaching depends on what other people think,” Ball told me, “not what you think.”

The idea that just knowing math was not enough to teach it seemed legitimate, but Ball wanted to test her theory. Working with Hill, the Harvard professor, and another colleague, she developed a multiple-choice test for teachers. The test included questions about common math, like whether zero is odd or even (it’s even), as well as questions evaluating the part of M.K.T. that is special to teachers. Hill then cross-referenced teachers’ results with their students’ test scores. The results were impressive: students whose teacher got an above-average M.K.T. score learned about three more weeks of material over the course of a year than those whose teacher had an average score, a boost equivalent to that of coming from a middle-class family rather than a working-class one. The finding is especially powerful given how few properties of teachers can be shown to directly affect student learning. Looking at data from New York City teachers in 2006 and 2007, a team of economists found many factors that did not predict whether their students learned successfully. One of two that were more promising: the teacher’s score on the M.K.T. test, which they took as part of a survey compiled for the study. (Another, slightly less powerful factor was the selectivity of the college a teacher attended as an undergraduate.)