Thursday, June 14, 2007
What Does Liping Ma REALLY say?
I saw Liping Ma speak at Rutgers University on July 19th, 2000. I
took extensive notes during her talk, and in thinking about the many
misrepresentations I have read of her actual thinking about
mathematics education, both here and in Mainland China, I decided it
was time to re-read KNOWING AND TEACHING ELEMENTARY MATHEMATICS, the
book that shot her into the limelight in the midst of the on-going
fight over what mathematics teaching and learning is about. It has
disturbed me to no end to see here balanced and reasonable ideas
distorted and "reinvented" and misused to justify what is usually
referred to as "traditional" math teaching in the United States, both
by activists in national groups like Mathematically Correct and NYC-
HOLD, and on local anti-reform blogs and discussion lists such as
Linda Moran's beyond-terc and many others. One would think that
Professor Ma had at some point signed a secret accord with such
folks, but my best guess is that she remains now as she appeared to
be in that summer of 2000: doing her own thing based on her
understanding of what matters, outside the usual politicized fray.
Given that she is one of the three people on the current National
Math Panel who is both qualified to speak intelligently about both
mathematics AND mathematic teaching and learning in elementary
schools (the other two being Deborah Ball and current NCTM president,
Francis "Skip" Fennell, I really wanted to refresh my understanding
of her views.
Thus, I was thrilled to discover that I had saved those lecture notes
from 2000. As a service to those who actually want to know what she
was thinking at that time, quite shortly after her book appeared, I
am going to reproduce those notes here. Of course, it is unlikely
that my doing so will sway those with entrenched beliefs that require
them to filter out anything that might cause cognitive dissonance.
But getting this on-line into publicly-accessible form will be a
benefit for many others who find it.
The talk began at 1 PM on July 19th, 2000 and was held at Rutgers
University in New Brunswick, NJ. It was a public lecture. If memory
serves, there were probably about 100 people or so in attendance.
Among them were teachers from School #2 in Paterson, NJ, who have
been participating in a long research project involving the use of
Japanese-style lesson study as a means to improve the quality of
mathematics teaching and learning in a multi-racial urban district.
This caught my attention at the time because my late uncle, who held
a Master's degree in mathematics from Columbia University, taught
industrial arts at School #2 until he retired in the mid-1970s. (He
was never allowed to teach mathematics in New York City, where he
originally sought work, because of his heavy Russian accent. In the
1930s and '40s, an oral examination was required of prospective
teachers, and it appears that accents could be used to keep otherwise
qualified teachers from working in their areas of expertise, an irony
given Professor Ma's accent as well as those of many people now
teaching throughout the country). Also attending were several
professors of mathematics education from Rutgers, as well as Fran
Curcio from Queens College in NYC, and Patsy Wang-Iverson, a Senior
Scientist with Research for Better Schools. Her primary
responsibilities, under the Mid-Atlantic Eisenhower Consortium for
Mathematics and Science Education, include overall management of the
Third International Mathematics and Science Study (TIMSS)
dissemination throughout the Mid-Atlantic region. Dr. Wang-Iverson
was responsible for the Paterson teachers being at this public talk.
Here are my notes, slightly edited for clarity or to insert questions
where I'm not sure what I have written down in terms of whether Prof.
Ma made the point or I was jotting down parenthetical questions. For
the most part, however, these are her thoughts and ideas as presented
during that hour lecture.
Introducing herself, Liping Ma told us that she taught elementary
mathematics in China for seven years and was later a principal
there. She came to the US to get her doctorate in mathematics
education, first at Michigan State where she worked with Deborah
Ball, Maggie Lampert and others, and later at Stanford, to which she
transferred because here husband didn't enjoy Michigan weather. She
finished her Ph.D at Stanford with Lee Shulman.
Ma states: We can't change US culture. We can't change the language
[does she mean the language Americans use to think and speak about
math education?] We CAN change teachers' content knowledge.
US has more resources to change teachers' knowledge (than does
China). In China, to give student a cup of knowledge, the teacher
needs a bucketful of knowledge. Ma didn't find information here on
what constitutes "good understanding," e.g., what is supposed to be
in each teacher's bucket?
Her research was with teachers from five Chinese schools. They
included some "best" Chinese teachers. Three of the schools were in
Shanghai and included one school considered to be a "best" school.
Also, one school from a small town, and one rurual school (her own
former school). She had 72 teachers in her sample.
The US data was drawn from research done at MSU by Deborah Ball. Ma
conducted no interviews or original research on US teachers. One of
the groups in the US sample was at the end of their first year of
teaching, earning Master's degrees at MSU at the same time. Others
were "above-average" and experienced teachers [no information offered
on what criteria went into determining their standing as "above
The terms "decomposing a higher value unit" and "composing a higher
value unit" in Chinese come from abacus terminology [technology rears
its head in unexpected ways]
Just two syllables: "jin yi"
One reason Chinese teachers have richer base of forms of regrouping
is because they let the kids give their ways and then discuss those
ways, including where it makes sense to use those various ways. And
the profound grounding "within 10" and "within 20" makes the kids
have lots of ideas, so there is an interactive process between "old"
teaching classroom practice (discussions), and what teachers have
available to draw upon for teaching other related topics.
N.B. : Ma only addresses subtraction and multiplication given the one
hour public lecture format. With about 15 minutes left, she chooses
to look at the multiplication issue rather than the division of
fractions or geometry: these are briefly mentioned only during the
introduction about the Ball questionnaire.
Chinese teachers see similar errors to those in the US, but address
them at the 2 digit times 2 digit level and try to get at the
misconceptions even at the 1 digit x 1 digit level by asking about
what units are being dealt with (e.g., 5 x 2 = 10 what? 10 'ones')
[Not 100 % sure if this following is her comment or mine, but I think
she said it: "It's wrong to say that US teachers 'only' want kids to
understand what the teachers understand themselves, which, generally,
is strictly procedural; they are no different from Chinese teachers
in what they DESIRE; they only differ in the DEPTH of their own
knowledge. How can someone want something for kids that s/he doesn't
realize even EXISTS?]
Ma says she has her students "invent/discover" how to do three-digit
multiplication and has those who succeed teach the other students how
to do it.
The seeds of advanced math are embedded in "elementary" mathematics;
the more solidly students and future teachers know arithmetic, the
easier higher math will be for them. But teachers of elementary math
don't need advanced math to teach elementary math effectively.
Stigler and Stevenson claim that US teachers teach "too fast." Ma
claims that Chinese teachers in reality only go slowly on key topics,
Why do Chinese teachers grow in knowledge from their teaching
experience, while US teachers don't? Chinese teachers don't assume
you (new teachers) know the math [?] when you start, but study and
grow and learn from colleagues and books.
Following this was a Q & A session:
Ma: In China, they've learned from some US/Western practices. For
example, kids sit two to a desk for pair work, and two groups of two
(two desks) for small group work. Also do larger group work.
First grade students engage in classroom discussion, always given
time limits for tasks; e.g., use 8 sticks to make figures and write
number sentences ot describe your figure. For example, 1 + 3 + 3 + 1
Kids tell stories about the number 8 (for example) based on what they
discover from these activities.
As a result, they really know composing and decomposing small
numbers. By the end of first grade kids know all the multiples up to
5. In China, especially in the cities, teachers are specialists and
teach only math in elementary grades (Ma's caveat: in Taiwan,
Singapore, Japan, the elementary teachers are NOT specialists and
still do a very good job on their math teaching). Kids spend about
45 minutes a day on math. Textbooks are small (shows the Japanese
textbooks for six-year-olds: about 3 inches wide in TOTAL. Each book
is about 1/2 to 3/4 inches thick.) In Japan, kids have their own sets
of manipulatives, unlike in China, and the quality of them is better
than the ones teachers have available in China. Chinese kids share w/
classmates. Teachers who have PUFM (profound understanding of
fundamental mathematics) believe they can teach effectively within
the 45 minute structure except for the lowest-level kids. In China,
drill is much better designed than in the US. There isn't "drill for
drill's sake," but rather based on understanding. In California, Ma's
son gets drilled on the same material 2 or 3 times a week. She thinks
badly of this practice. In China, drill "moves up" [not sure exactly
what she means by this].
In China, teachers often concentrate on the students' textbook and
feel that the teachers' book is mostly useless. The teachers prefer
to interpret the student book themselves. They ask each other why an
example is given by the author and they evaluate it as useful and
connected, or not. They meet once a week (usually by grade level, not
subject) to discuss their teaching. Most teachers know at least a
"small loop" of topics: e.g., 1st & 2nd grade topics or 1st, 2nd, and
3rd grade topics. Some (those who have PUFM) know grade 1-5 topics
well, and they get to teach all five grades.
Teacher guides don't generally include anticipated student errors &
typical responses, but teachers share this information with each
other from their practices. Ma was given the chance to spend a month
in some good classrooms when she started out, to observe and learn
from good teachers. But the same principal who allowed this trained
another teacher by noting and analyzing EVERY student error made in
that teacher's classroom. So there are varied approaches for
different teachers possible.
The term "knowledge package" is Ma's, not something generally used in
China. In China, the principal decides, mostly based on his/her
immediate needs, what new teachers will teach. If the principal
believes a teacher is "promising," that teacher gets to loop more
among the different grades and becomes what is termed a "backbone"
teacher for that school.
When asked how she "corrects" her son's education in California: I
teach him the 'most important concepts.' You can learn from "what you
already know. Simple things." Her son isn't tops in speed, but she
says she feels that is fine. Someone else asks a question that
expresses negative views on calculator use in elementary math
classrooms: isn't lack of PUFM 'masked' by calculator use? Ma says
her son learned negative numbers not from his older sister but from
playing with a calculator. Ma tested him and found his understanding
of negative numbers to be good. She thinks it was rooted in his
number sense. She believes we need much more research before saying
"yes" or "no" to calculators. In China, there is no official national
policy on them.
Asked why America has such resistance to cooperative and
collaborative work among TEACHERS, Ma gives a very diplomatic answer:
"I'm a foreigner and don't have a why."
Asked, "what are the Big Ideas of elementary mathematics, and where
does one find this information? Ma says she's working on this as her
next project but warns that she's not read that much of the literature.
The teacher day in Chinese elementary schools is typically starts at
7:30, with classes for the kids ending around 2:30 or 3. Teachers
teach six periods, but many run math clubs and groups for slower
kids, so teachers often leave at 5 pm or later. Teachers are expected
to grade every problem and the kids are expected to correct every
error. The school year is 42 weeks
Very few kids are held back; the teachers design their own tests and
Blind spots in American mathematics education: ordinality is taught,
but not cardinality (or not sufficiently so), and that is one of the
important Big Ideas. American teachers over-stress counting-on and
counting back on the number line, as opposed to seeing numbers as
sets [uh, oh!].
Ma is worried about using concept maps and writing to assess students
specifically because the quality of the teaching of writing here
isn't strong enough.
Q: Should students be interviewed on their knowledge (like in the
Ball data, and Ma's use of TELT (?) questions)? Ma: hard to get the
right people as interviewers. She thinks you can know without
interviews what the kids know [she doesn't elaborate on this].
Q: How important is teacher content knowledge?
Ma: You cannot separate teacher content knowledge from the teachers'
knowledge of math pedagogy and and their knowledge of how kids learn
math. They must know what something in math is, how to do it, and
what it means.
Q: About division: when is this taught in China and why isn't it so
hard for Chinese students (as opposed to Americans)?
Ma: This is rooted in linking division to multiplication as its
inverse. e.g., 5 times 3 means 3 groups of 5 elements each. it is
important to stress the partitive and quotitive models early, as is
done in China by late 3rd grade or early 4th grade). Skip over the
"circle" idea (?) quickly and go to fractions as another model for
Q. About limits on what numbers are explored, e.g., numbers to 9,
within 20, etc.: what about more open-ended exploration?
Ma: I've been thinking a lot about that. Many Chinese teachers do
lots of discussion. Teachers put various methods of, say, regrouping
on the board and ask kids why we have them and why they and
conventional ways are used.
Q. What is the relationship between advanced math classes for
teachers and their ability to teach mathematics?
Ma: Better to do deep exploration in classrooms of elementary
"Decomposing a higher value unit" is Ma's term. In Chinese, it's more
like "stepping back" on an abacus.
I asked: Could you compare and contrast the Ball and Lampert
classrooms, based on the available videos and other data from their
1989-90 teaching of grades 3 and 5, respectively, at Spartan Village
Elementary School in East Lansing, with their discussions over
several days on a central problem of the day which kids explore and
work on, with what you've seen in Chinese classrooms?
Ma: I looked at the [very well-known] Shea numbers video and showed
this to teachers in China. The teachers there with good PUFM really
liked it, but felt that given constraints there, they could only "go
wild" like that a few times each year.
Ma: A good textbook is the most important thing we can offer teachers
to improve their knowledge of mathematics. But 90% of teachers here
believe they already know enough elementary math. So a good textbook
would need to be able to show them that they are more ignorant than
they believe and then help to reduce that ignorance.
She feels that as far as student materials go, we need to develop our
own textbooks for American students.
Patsy Wang-Iverson: In the US, kids try to figure out what the
teacher is thinking (and then do that). In Japan, the teachers try to
figure out what the students are thinking so that they can help the
students make more sense of the mathematics.
In China, there is no tracking or ability grouping until 10th grade.
Ma: "China (and EVERY country) has serious problems in its public