In November of last year, 200 people, mostly mathematicians, sent an open letter to Secretary of Education Richard Riley. The letter asked him to withdraw the labels "exemplary" and "promising" that the Education Department had recently applied to 10 innovative math programs. I do not know the validity of the criticisms. I do know that the last time the nation let mathematicians develop K-12 curricula, in the post-Sputnik panic, the result was a debacle known as New Math.

I mention the letter to illustrate that the teaching of mathematics has become a politicized matter, just as the teaching of reading has been. Two recent books help us see why. Simply put, something has to be done to improve the mathematics knowledge of America's teachers. In *The Teaching Gap: Best Ideas from the World's Teachers for Improving Education in the Classroom*, James Stigler of UCLA and James Hiebert of the University of Delaware analyze videotapes of middle school instructors teaching mathematics in Japan, Germany, and the United States. These tapes were made as part of the 41-country Third International Mathematics and Science Study. *Knowing and Teaching Elementary Mathematics* is a study of how math is taught in elementary schools in the United States and the People's Republic of China, conducted by Liping Ma at the University of California, Berkeley.

Neither book has graceful prose, but neither book is obtuse or difficult, either. *The Teaching Gap* is written for the educated general reader; *Knowing and Teaching Elementary Mathematics* is for the kind of people who delve into *Scientific American*. Both have important things to say about teaching mathematics in this country.

From the cover through the initial chapters, I distrusted *The Teaching Gap*. The title plays off *The Learning Gap*, an earlier book co-authored by Stigler. But *The Teaching Gap* in no way compares "best ideas" from the "world's teachers," as its subtitle suggests it does. There is not one such idea in the entire volume. Even if there were, the study would have no way of directly linking those ideas or teaching practices to mathematics achievement.

The inside flap of the cover contains outright errors. It opens with, "For years our schools and children have lagged behind international standards in reading, arithmetic, and most other areas of achievement." This is not true. In the most recent international comparison of reading, only one nation out of 31 scored significantly higher than American students. And in the only study that had "arithmetic" as a category, American students were average. In the science segment of the Third International Mathematics and Science Study, American students were above average.

The authors repeat a claim from an earlier study that "the highest-scoring classroom in the U.S. sample did not perform as well as the lowest-scoring section in the Japanese sample." The data from the latest mathematics and science study make it clear that this earlier finding occurred because the Japanese sample was not representative of the nation. While Japanese students scored substantially better than American kids, there was also substantial overlap between the scores of the two groups. Moreover, American students in a group of suburban school districts scored almost as high as the Japanese students, getting 70 percent of the items correct, compared to Japan's 73 percent. Stigler and Hiebert would have done better to leave aside previous research. Their study stands on its own.

For Stigler and Hiebert's book, a team of six people, two from each country, coded the Third International Mathematics and Science Study videotapes. The description of the code development process does not exactly inspire confidence. "The discussion was so vigorous that it often would take a day or more to get through a single lesson. There were disagreements in the group about the content of the tapes, and especially about how to describe them." It would be nice to know more details. It would have been even better if another team had also developed a system of analysis independently. Would it have contained the same elements?

The team found that American teachers present about twice as many definitions as Japanese and German teachers. This accords with other findings in the mathematics and science study that American textbooks are about three times as thick as those of other nations and that our teachers try to teach it all, often covering topics only briefly and shallowly. Stigler and Hiebert discovered that American lessons were devoid of mathematical proofs. About 10 percent of German lessons and more than half of the Japanese lessons contained such proofs. American teachers stated concepts, but did not develop them. Only 22 percent of the lessons in this country contained developed topics, compared to 77 percent in Germany and 83 percent in Japan. Not only did German and Japanese teachers develop topics; they linked them to other topics. When lessons were rated on the interrelatedness of their parts, German lessons scored four times as high as American lessons, Japanese lessons six times as high.

Japanese and American teachers organized their lessons quite differently. In a typical American lesson, a teacher reviewed homework, demonstrated how to solve the problem of the day, gave students classroom practice, corrected that work, and assigned homework. Japanese teachers reviewed the previous lesson, presented the problem of the day, and set the students to working on its solution either individually or in groups. The class then discussed problem solutions (some problems had more than one), often led from the blackboard by students who thought they had successfully solved the problem. American students almost never led such a discussion.

American and Japanese teachers organize their lessons differently in large part because they believe different things about what mathematics is and how to teach it. American teachers believe that mathematics is a set of procedures; they want their students to become skilled at these procedures. Japanese teachers, by contrast, "act as if mathematics is a set of relationships between concepts, facts, and procedures. Japanese teachers wanted their students to think about these relationships in new ways."

These differences seem profound, but as I noted at the outset, there is no way to relate any of them to the differences in performance. Indeed, one is left with an enigma: On many dimensions, the German teachers are similar to Japanese teachers, yet while Japanese students attained much better scores than American students, German students scored the same as Americans.

It could well be that the Japanese students score higher because so many of them attend *juku*--cram schools--after school and on weekends. *Juku* specialize in teaching students how to take tests. *The Teaching Gap* does not discuss the role of *juku*. There are other possible explanations. A Japanese educator who, at my request, watched some mathematics and science study tapes, concluded that, indeed, the Japanese classes were more conceptually oriented. He felt, though, that Japanese teachers are free to teach conceptually because they can count on family support to ensure that less glamorous mathematics activities will be completed at home. American teachers cannot count on such support.

While Stigler and Hiebert analyze teaching, Liping Ma presents teachers with problems in mathematics instruction and asks American and Chinese teachers how they would approach the problems. For instance, how would they teach subtraction with regrouping? If, when multiplying two three-digit numbers, their students did not line up the three mini-products correctly, how would they explain what to do and why?

Ma's American teachers sound very much like Stigler and Hiebert's, and her Chinese group closely parallels the Japanese. In the subtraction with regrouping problem, for instance, Ma writes, "Seventy percent of the U.S. teachers and 14 percent of the Chinese teachers displayed only procedural knowledge of the topic. Their understanding was limited to surface aspects of the algorithm."

American teachers knew what needed to be done, but knew only the procedure, the algorithm. Some American teachers could not explain why each mini-product in the multiplication of two three-digit numbers was moved one digit to the left: "I can't remember that rule. I can't remember why you do that. It's just like when I was taught, you just do it."

Ninety-two percent of the Chinese teachers showed a conceptual understanding of the problem, explaining it in terms of the place value of the different columns and the distributive law of mathematics. American teachers who mentioned the "tens" or "hundreds" column seemed to use those words only as labels.

When Ma asked teachers to divide one and three-fourths by one-half, only 43 percent of the American teachers gave a complete and correct answer. Another 9 percent applied the algorithm properly but then did not reduce the answer and convert to a proper fraction. Some multiplied where they should have divided. Some divided by two, not by one-half. Not only could most Chinese solve the problem and explain it conceptually; some offered alternatives to the most common form of solution.

Ma reports that while many American teachers said they wanted to "teach for understanding," their limited knowledge of the problems would prevent such teaching. In all of the problem settings, Chinese teachers related the problem at hand to other mathematical concepts while American teachers considered the problem in isolation from anything else.

The notion of finding Chinese and Japanese teachers who are "comparable" to the American group is probably not meaningful. But if one pays attention only to the Americans in both studies, one is struck by the similarity of the findings. American teachers emphasize rules, procedures, and algorithms, and in some instances see no need to go beyond such knowledge, or have no capacity to. Both books emphasize that better mathematics education must put teaching competence--particularly competence that develops after the teacher starts teaching--front and center.

One important contribution of these books is to show that American classrooms have not been taken prisoner by the acolytes of Rousseau and Dewey. In his book *The Schools We Need and Why We Don't Have Them*, E. D. Hirsch, Jr., argued that just such a capture has occurred, leaving our schools to fail at the hands of these antiknowledge "romantic progressives." The Third International Mathematics and Science Study tapes and Ma's findings indicate that, if anything, it is American teachers who are obsessed with getting knowledge out of their heads and into the kids' noggins in a most traditional, fact-based approach.

Susan Ohanian's wise little book *One Size Fits Few* is a look at the folly of "Standardistos." These are people who seem to have little interest in the dynamics of classrooms and the needs of the kids themselves. "How is all this going to work?" a parent asked former Secretary of Education William Bennett after a speech on the need for tougher standards. "I deal with wholesale; you're going to have to work out the retail for yourself," Bennett replied. "There you have it," writes Ohanian. "Standardistos don't give a damn about how their plans and panaceas might work in classrooms."

Lest someone think Ohanian is exaggerating the prevalence of this kind of thinking, she fills her book with quotes from Standardistos par excellence such as the late Albert Shanker, who was president of the American Federation of Teachers. Shanker contended, "Unless we have standards that tell us, grade by grade, what the teacher is required to teach and the student required to learn, many of our students will not reach the level of competence that we expect of high school graduates."

Nonsense, Ohanian argues. Against such pronouncements, she offers counterexamples: individual, idiosyncratic kids and successful adults who followed their own unusual paths to success. A longtime teacher, she also objects to the very idea of expecting all students to study and master the same thing. She notes that her third-grade students encountered Aesop and Robert Louis Stevenson, two worthies who made E.D. Hirsch's list of approved topics and people. But, she goes on to say, they also encountered Jean de La Fontaine, Basho, Langston Hughes, Laura Ingalls Wilder, and E.B. White, who didn't. Will her students therefore fail to attain the "level of competence" of which Shanker spoke? That's for Standardistos, not real teachers (or parents or kids), to worry about. And they do. In a late 1999 speech, Hirsch declared, "A classroom of 25 or 30 students cannot move forward until all students have gained the knowledge necessary to 'getting' the next step in learning."

Such people believe that if the standards are there and teachers are trained to strictly adhere to how they should be taught, kids will master the material--and teachers will become a mere "variable." University of Texas Standardista Professor Barbara Foorman says, "The teacher variable does not contribute significantly above and beyond the curriculum, so what we have here is a powerful mathematical model. My hypothesis is that the teacher variable will be even less significant within the direct instruction group."

Against the widely held tenet that today's students don't know very much, Ohanian levels the charge that we are asking them to know more and more complex things earlier than ever. We are putting them under tremendous pressure. No wonder the use of Ritalin and Luvox is increasing.

My experience corroborates Ohanian's assertions. Consider two examples.

An October 1999 *Washington Post* article mentioned casually that a sixth-grade class used Venn diagrams to solve problems. I first encountered Venn diagrams in a logic course as a college junior. So did the reporter. When I mentioned this to a group of teachers, some said they were using them in third grade.

A 10th-grade social studies standard in Virginia's Standards of Learning program reads thus: "The student will analyze the regional development of Asia, Africa, the Middle East, Latin America and the Caribbean in terms of physical and cultural characteristics and historical evolution from 1000 A.D. to the present."

This second example is not extreme or taken out of context; nor is it unique to Virginia. Fifth-graders in California must memorize the periodic table, and sixth-graders in South Dakota will know more about ancient Greece than most college graduates who major in ancient history.

One indication that something is amiss with the standards movement can be seen in a January 2000 report from the Thomas B. Fordham Foundation evaluating the standards of the various states. Oddly, states that had the best standards, according to the evaluation, scored low on domestic achievement tests and international comparisons. States whose standards were labeled "lousy" scored high on both types of assessments.

Perhaps the answer is for teachers and principals to take a Standardisto to school. Ohanian gives us the case of Robert Wycoff, president emeritus of ARCO, who spent just one day shadowing the principal of a high school. He stunned reporters by praising the principal, teachers, and kids. "If I were principal of Manual Arts, I could not do as good a job as I saw today. This is not as simple as going to the moon."

It's not. The laws of physics are known. The laws of kids are not. You can aim a rocket with precision because all variables can be either controlled or factored in. Kids bring active minds and wills to the scene and will thwart, surprise, and delight you with their unpredictability. Ohanian urges us to celebrate that variability and spontaneity.

This is a lively, funny, and angry book full of stilettos and wry barbs. It is a perfect antidote for the reactionary educational times we live in, a time when a *USA TODAY* cartoon depicts a mother reading to her child in bed and saying, "The little pig with higher verbal and math lived happily ever. The other two pigs were swallowed by the wolf."

Early on, Ohanian invokes *Network* and suggests teachers should rise up yelling, "I'm mad as hell," and refuse to teach the standards or give the tests. That day might come. Rebellions are brewing all over the country, not just among teachers but among parents and students as well. Perhaps we can eventually return to a time when, in the words of Philip Lopate, "the art of teaching is knowing when a student is best left alone and when he is ripe to receive your help." It's not the same time for every child. ¤