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Tuesday, March 21, 2017

Most American Math Teachers Cannot Teach Math

...because they studied non-math in school, not math. (And most of the rest of us have the same problem.)

I read Dr. Nancy Pine's book, Educating Young Giants, with great interest. The book is about her observation of classes in China, her discussions with Chinese teachers and parents (mostly through interpreters), and the comparisons she makes to American education. She admits to being ethnocentric at the time of her first visit to China in 1989, but while she could sometimes recognize her own egocentricity, she was not able to fully overcome it.

She noticed that in Chinese literature classes, teachers emphasized close reading and digging for the author's meaning. She felt that Chinese teachers denied students the opportunity to create personal meaning from the literature they read. Although her research in China centered on elementary literacy development, as a former math major (page 41), she became interested in observing elementary math classes. As everyone who has ever observed Chinese math classes has reported (see, for a few of many examples, Harold Stevenson, James Stigler, Liping Ma), she, too, witnessed superior teaching skill.

I have been teaching math in China for the last several years, and I taught in Japan for nearly two decades. I speak both Japanese and Mandarin. My conversations with people from mainland China, Hong Kong and Taiwan, as well as written descriptions such as Educating Young Giants has led me to conclude that the actual education systems as well as the cultural foundations of both China and Japan are very similar.

Nancy Pine came to appreciate that Chinese teachers teach mathematics, but “most U.S. teachers merely teach arithmetic” (page 45). Dr. Pine is being generous. U.S. teachers teach non-math, specifically routines, tricks and shortcuts, but call it math on the misconception that if numbers are running around, it must be math.

Chinese teachers spend a significant amount of time considering a relatively simple math problem from every conceivable angle. The students probably already know the “answer” and that is precisely the advantage of using an easy problem. Because they already know the outcome, they can concentrate on the process, the concept-building. Once the concept is solid, their homework includes problems that American students eventually spiral to. China thereby reduces the need for the endless review so common in America.*

Dr. Pine herself admitted “that even with my strong interest in math, I would not have known enough about the underlying mathematical concepts to think through the best ways to present the initial problem that would enable students to correctly solve more complex ones” (page 45). See what she is saying? She is admitting that she was great at non-math, but weak at mathematics itself. Not only that, she says she knows “that most American grade-school teachers, who teach five or more subjects, do not have the depth of knowledge to walk children through mathematical concepts to prevent misunderstandings” (page 50).

She believes it is because American teachers are generalists who must teach every subject, while Chinese teachers are specialists who teach only one subject. I would like to suggest that being a generalist or a specialist has nothing to do with it. Chinese teachers could be generalists and their ability to teach math would still “far surpass ours” (page 46) because nearly all Chinese teachers, regardless of their particular specialty, acquired a profound understanding of fundamental mathematics (PUFM, a term coined by Liping Ma) beginning in the primary grades. If our own children acquired PUFM, they would also be much more effective math teachers, even as generalists.

You see, regardless of professional training or subject matter courses, teachers tend to teach the way they were taught. The strident calls for teachers to take more subject matter courses is misplaced. Simply learning more and more non-math will not improve teaching ability. Okay, how about we reteach math at the university level? I tried to do exactly that, only to meet with terrible resistance. “We don't want to know why the math works,” my students complained, “Just tell us how to get the answer.” Fine, let's at least teach those students who aspire to become elementary teachers. Guess what? Most universities require all elementary teaching candidates to pass a series of courses entitled something like “math for elementary teachers.” My students complained that the classes were a waste of their time, since they “had learned all that stuff in elementary school.” Most elementary teachers, even though compelled to take a real math class, most of them for the first time in their lives, end up graduating from college without learning much math due to their resistance. They subsequently teach math the way they were taught in elementary school.

The main reason that Chinese students do so well in international math tests is because they actually learn math in school. American do not. Therefore, the reasons critics cite (specially selected students, lower poverty rates, rote learning, etc) miss the point. What critics are saying is that due to circumstances beyond our control, American students can never compete with Chinese students. I call baloney. If we would actually teach math in our schools, our students could compete just fine.

Next: examples of non-math teaching I encountered in a child's algebra class.

After seeming to correctly solve a number of simplification problems of the form -(ax-b) or -(ax+b), a child complained she could not simplify this one: +(ax+b). “What do I do with all those plus signs?” she wailed. What did you do with the other ones? I flipped them (referring to a mat-and-tile manipulative she is using in class). Even after all that flipping, she still had no idea what was going on. As long as she flips correctly, she can get the right answer without ever understanding how the flipping was supposed to communicate the concept. (Here is another topic: how American teachers routine take great resources like manipulatives and use them ineffectively

In another example, the child needed to solve for x by first combining +2 + ¼, easy—the answer is 2¼. But the next problem was +3 –  -½. She wrote 3-½ as her answer, then complained because the problem was coming out “all weird.” I straightened that one out with her, only to have her evaluate the next problem, -2+ 2/3, as -2 2/3.

Dr. Pine realized that the depth of Chinese math learning far surpassed ours. Yet she seemed unable to perceive that the digging for meaning she observed in literacy classes was precisely the same digging for meaning evident in math classes*. She lauded it in math but lamented it in literacy, saying that teachers denied Chinese children the expression of their own personal opinions.

* Everywhere I wrote an asterisk I am referring to the Chinese philosophy of math education as evidenced by the textbook presentation of concepts and the implementation I and others have observed  in many Chinese classrooms.  HOWEVER, honesty compels me to relate that there are a number of  Chinese math teachers whose delivery of math concepts is at best cursory.  These teachers also assign an overwhelming amount of homework and "practice tests."  These teachers have been know to "steal" time for "unimportant" subjects like art and music for more practice tests in their never-ending quest to maximize test scores regardless of understanding. This approach is absolutely murderous to the spirit and curiosity of Chinese students.