Many people think that good teaching is like pornography -- you know it when you see it. But is that true?

Suppose you walked into a high school math class that was obviously well-run. You could see there was a good relationship between the students and the teacher -- cordial and respectful -- so discipline was no problem. The teacher kept straight lecture to a minimum; instead of talking at the students he guided them to work the examples for themselves. And the students responded by paying attention and asking questions about what they didn't understand.

And suppose you found out that the teacher had done such a good job preparing his students for a statewide exam that they had scored in the top 15 percent. Would you say that you'd seen good teaching? Sure. But Alan Schoenfeld, a professor of mathematics education at the University of California at Berkeley, says you'd be dead wrong -- at least if you think that good teaching means
helping students gain a real grasp of the subject and solve real problems.

In fact, Schoenfeld, who observed this plane geometry class in a New York State high school over an entire school year, found that this great-looking math class encouraged practices and attitudes that gave students the wrong idea about mathematics -- and, indeed, might block any real understanding of the subject. He describes what he saw in an article provocatively titled "When Good Teaching Leads to Bad Results: The Disasters of 'Well-Taught' Mathematics Courses" (Educational Psychologist 23(2), 145-166, 1988).

So what was wrong? In order to get his students to succeed on the New York State Regents Examination, the teacher tailored his teaching to the test, presenting techniques for getting the right answers to test questions in the time allowed on the test. He stressed knowing the procedures cold -- so students could perform them almost without thinking. He had them practice using old Regents exams. And he coached them on their test-taking skills, emphasizing that they needed to work as quickly as possible and abandon any problem that they couldn't solve within 5 minutes or less. The teacher's advice clearly helped his students get good grades, but it also encouraged an unthinking approach to solving math problems -- though of course this was not the intention.

Students taught in this way, Schoenfeld says, figure that if they don't get an answer right away, they never will -- so they might as will give up. As a result, they don't learn "to engage in real mathematical thinking -- in trying to make progress on difficult problems, in engaging in the give-and-take of making sense of complex situations, in learning that some problems take time, hard work, and a bit of luck to solve."

What they do learn is that understanding math is not really important (and only geniuses can understand anything about math anyway). The important thing is to get the right answer -- even if you don't understand the problem. And the way to do that is to master the mechanical techniques the teacher has presented.

Was the teacher Schoenfeld observed at fault? Not really. He was doing what the system expected him to do. And, Schoenfeld says, the story would have been the same in most other classrooms in the country because the standard curriculum in K-12 mathematics and the standardized tests put a premium on the kinds of skills this teacher was pushing -- and offer little encouragement to think through real mathematical problems.

In fact, Schoenfeld's observations about the plane geometry class dovetail with examples from all over -- like the one about a group of 13-year-olds taking a National Assessment of Educational Progress (NAEP) exam who did a computation right but got the answer wrong.

The youngsters were asked how many army buses, each holding 36 people, it would take to transport 1,128 soldiers. Seventy percent of the students carried out the long division correctly. But 29 percent of them said the number of buses needed was 31, with 12 left over; and 18 percent rounded their answer to 31. In other words, less than a third of the students who could divide the two numbers knew they also needed to think about what the problem said. The 12 left-over soldiers were just a "remainder," rather than people needing another bus.

Schoenfeld's point is that what looks like good teaching can get students good marks without getting them to understand what they're doing. Teachers usually shape their teaching to a test, and the time pressure of tests and the shortcuts this pressure encourages also encourage fundamentally wrong attitudes towards math.

To most people, it seems easy. Teach the subject and then test students on it. But a test is not just a thermometer -- a way of diagnosing how much students have learned. It affects what's taught and the way it's taught. If a teacher knows that his students will have to get the correct answers for 20 problems in 60 minutes, that's what he'll teach his students to do. 

We'll go a long way toward getting better teaching -- and better learning -- when we have better tests.