A Wire-watching buddy tells me, “Man, I’d never teach in a city school. The eighth graders can’t even do fractions!” When I ask him why he thinks that is, he cites the usual litany of family and neighborhood challenges kids face in West Baltimore, the ones portrayed so vividly and heartbreakingly in The Wire itself. But the scene my friend was referring to illustrated what the real culprit is: Our schools often do a terrible job teaching math. When one of Prez’s students says he can’t complete the fractions worksheet because “we never did one-thirds,” Prez responds, “One-fourth, one-fifth, one-third: Follow the same steps.” He’s right in one sense: Certain steps will produce correct answers. But he’s doing nothing to address the much larger problem: Many of his students clearly have no grasp of what fractions are or what “doing arithmetic” with them really means. They've studied a set of steps, but they don't understand how those steps apply to different denominators or problems that don't involve food items.
To illustrate, let’s assume they were dividing fractions, e.g. problems like “10 / ½ =”. Most of the kids might recall from studying division by whole numbers that when you divide, you get a smaller number. Their elementary school teachers might even have encouraged them to check their work by looking to make sure their answer is smaller than the number they started with. But when you divide by a fraction, you get a bigger number, in this case 20. Instead of working with students to help them understand why, many teachers simply drill them in a rote, two-step procedure for dividing by fractions: “invert and multiply.” Students are asked to memorize the word “invert” and told that it means “to flip,” and then asked to memorize and practice those two steps. Sure, that produces the correct answer, but simply memorizing a procedure does little to develop real mathematical understanding. They’ve simply been drilled to follow a seemingly arbitrary set of steps that produces a counterintuitive result. No wonder they’re confused!
Math doesn’t have to be taught that way. In 1999 a young researcher named Liping Ma published a book that caused a huge stir in math education circles (though, sadly, very little buzz in the education policy arena). She found that Chinese teachers help their elementary students develop a much deeper conceptual understanding of math and offer them a broader repertoire of strategies for solving problems than do their American counterparts. That's partly because American teachers themselves tend to have a much shallower grasp of math concepts than Chinese teachers, despite spending more years in formal education to become a teacher. In fact, fewer than half of the American teachers in Ma’s study fully and accurately answered the problem “1¾ / ½ =”. It wasn't simply a matter of forgetting the steps but also a lack of conceptual understanding about what it means to divide by a fraction. Some teachers told her they divided 1¾ by 2 because they understood the problem as asking them to “divide something in half” rather than to figure out, say, how many halves there are in 1¾. One teacher admitted, “I can't really think of what dividing by a half means.” (Ma’s sample included only 21 American teachers, but others who have replicated her research with a larger numbers of teachers have found similar results.)
Ma’s book also sheds light on another perceptive element of that Wire scene. She found that poor conceptual understanding made it difficult for many American teachers to find helpful and accurate ways to represent fractions. U.S. teachers mainly used either food or money to represent fractions, while “those used by the Chinese teachers were much more diverse” and included many examples students would be familiar with from their daily lives, “such as what happens in a farm, in a factory, in a family, etc.” Recall that another of Prez’s students tells him she can’t do the fractions problems on the worksheet because they involve cars: “All that stuff we did in practice was about food!” Again, Prez's response isn't very helpful. He tells her it doesn't matter and to just “pay attention to the number.” Huh?
Such difficulties are not limited merely to teaching fractions in the upper elementary and middle grades; Ma documents how poor conceptual understanding can impede instruction in something as simple as subtracting two-digit whole numbers (e.g., 53 - 26 =). In fact, a hugely important study published last year (also summarized and discussed here) by several University of Michigan researchers found that teachers’ mathematical knowledge has a big impact on how much their students learn over the course of a year—even at the first grade level. They also documented that disadvantaged students—particularly minority youngsters—are more likely to have teachers with lower levels of mathematical knowledge and understanding, a finding they call “shameful.”
Although the full breadth of this problem and its implications have been largely ignored in policy circles, that might change soon. Liping Ma and Deborah Lowenberg Ball, one of the U. Michigan researchers, are both serving on the new, high-profile National Mathematics Advisory Panel. But getting traction won't be easy: This is an uncomfortable topic for many people, including teachers, who sometimes feel it amounts to “teacher bashing.”* Let's be clear: It’s the system that should come under fire. Teachers are themselves the products of the same shallow elementary math instruction they pass on to their students, and deficiencies in their mathematical understanding seldom get addressed later on, either during their time in ed schools or after they start teaching. In contrast, Ma says that Chinese teachers get lots of opportunities to build their math knowledge over the course of their careers. The good news? Another study by Ball demonstrated that “teachers can learn math for elementary school teaching in the context of a single professional development program.”
Of course, that would require a system that works thoughtfully to nurture knowledge and cultivate capacity. Ironically, says The Wire, Marlo and his colleagues are much better at doing those things in the system they run!
* Ball and a colleague discussed some of the negative feedback they’ve received for even conducting such research (e.g., testing teachers for research purposes is inherently wrong because it “de-professionalizes” them, etc.) in a must-read article for American Educator, the excellent magazine published by the American Federation of Teachers, last year.
--- Guestblogger Craig Jerald
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