...because
they studied non-math in school, not math. (And most of the rest of
us have the same problem.) https://schoolcrossing.blogspot.com/2007/11/are-you-good-at-non-math.html
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 https://schoolcrossing.blogspot.com/2010/11/i-love-math-manipulativesbut.html.
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.
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