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"Do not forget yourself as a teacher of yourself" - Liping Ma
Posted:
Jul 18, 2002 10:29 PM
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An Interview with Liping Ma
"Do not forget yourself as a teacher of yourself"
http://www.enc.org/focus/content/document.shtm?input=FOC-002866-index
With her book, Knowing and Teaching Elementary Mathematics, Liping Ma
caused many to question the depth of teachers' understanding of the
subject. For more about Liping Ma, see the related article A Deeper
Look at Elementary Mathematics .
by Terese Herrera, ENC Instructional Resources
Liping, through your book we glimpse a much deeper, coherent picture
of arithmetic. Our students need that. Our teachers, I believe, are
hungry for it. How do we achieve it?
I don't know. I really don't know. I think that many should work on
this.
First, mathematicians who teach content courses in universities should
think about arithmetic in a deeper way. In earlier centuries, great
mathematicians devoted their whole lives to solving complex problems
through applied arithmetic. Today's mathematicians are working on
advanced branches of mathematics, but for those who teach core content
for future elementary math teachers, knowledge of advanced mathematics
is not enough.
In China, deep arithmetic knowledge is not owned by the
mathematicians. It is owned by teachers at the elementary school
level. It is not in a book; it is in their minds, and they pass the
knowledge on to the next generation. Here in the United States, I
personally don't see this body of knowledge in teachers' minds. I
don't know why. I hope someone can research and document American
teachers' knowledge of arithmetic. This would be very helpful for our
teachers.
Math educators need to think not only about how teachers will teach
math but also about the math they are going to teach. In the United
States, one group of professors teaches content; another group teaches
method. But teachers realize that we cannot separate content from
method--what to teach from how to teach. In a teacher's real life,
they are combined.
We (teacher educators) have no way to put that deep knowledge of
elementary math in teachers' minds either. We don't have an answer to
give to them. Teachers should know that this is not something that
they should wait for other people to give them because we don't have
it. We should tell them that it is not their fault at all, and we
should encourage them to deepen their knowledge of elementary math
through their own teaching.
Teachers themselves should ask "Why?" all the time. When they teach
multiplication to their students, they can think about why it works.
Then they will learn elementary math in a deeper way through their own
teaching of it.
Again, I don't think that I have the answer. I am in the process of
looking for answers. There will not be an easy answer for what we
should do or a quick solution. We should work together--and work
hard--mathematicians, math educators, and teachers.
Your book focuses on arithmetic. How do you see the teaching of
geometry in the early grades?
Again, for me it is still a question. At the K-6 stage I see
arithmetic as the main focus of elementary math, even though
elementary geometry should be included.
The geometry that we are teaching now in this country at the
elementary level does not connect tightly enough with arithmetic. We
teach children the names of shapes--rectangles, triangles, and so
on--but not the connection between those shapes. Two congruent
triangles can make a rectangle, for example; this is a connection
between those figures.
And we need to teach the connection between the geometric figures and
other mathematics. For example, we talk about the perimeter of a
rectangle as equal to length length width width. The perimeter also
equals (length width) 2.
Children could discuss why the formula can be written both ways and
which way it makes more sense, is more clear, more mathematically
advanced. Doing so, they connect this geometry piece to arithmetic and
with the way they think mathematically.
Is there a role for calculators in elementary math classes?
I've had a question in mind for a long time: Since there are
calculators that can solve all the skill problems in arithmetic, why
do we teach children arithmetic? It is very important to ask this
question. It challenges us to sit down and think carefully about what
we really want our children to learn.
I believe that arithmetic is much more than skills; it is a basic tool
to develop children's minds, their way of thinking. The question
becomes: What role does arithmetic play in the development of a child?
It is just like asking: What role does literacy/reading play in the
development of a child?
I am always asked about calculators in the classroom: Should they be
allowed? This is not the question. Calculators are already in use.
What we need to consider is how they should be used. In what way can
calculators help students learn more mathematics and learn it in a
better way?
Your book has influenced the math education community at many levels.
What are you working on now?
I am a visiting scholar at the Carnegie Foundation because I have a
post-doctoral grant for research. My research is on what I call the
cornerstone ideas of elementary math. I often hear the terms "core
ideas,""big ideas,""important ideas," meaning those ideas that hold
all the pieces of a subject. However, it seems that no one is doing
systematic research on what these ideas are, why they are central, and
how they contribute to the learning of elementary math.
I began last year with pilot studies in kindergarten classes. I
developed some teaching materials, including multimedia math games.
All of these games are designed for teaching math ideas. My questions
are: How do these games work with children? How do students learn? How
can teachers learn through teaching?
I go into the classroom to observe the instruction and note what is
wrong with my idea. I don't send in others to collect the data. I use
myself as the instrument of thinking, because of my early experience
in teaching math and observing others teach, and because of my focus
on how to create good math teachers. All of those experiences form my
vision and make me, myself, an instrument to collect and analyze data.
It is a very personal and interior kind of research.
If grants become available, I want to study math textbooks to
determine their central ideas and work further with more teachers. The
ideas proposed as central to elementary mathematics should then be
tested in classrooms, K through 6. We need to lay out the arguments
for why certain ideas are central in terms of the subject and in terms
of children's learning, and how these ideas can be taught.
I understand you've also written an intervention program for
intermediate grades?
Houghton-Mifflin wanted to develop a program for children in grades
four and up who are below their grade level in math. Cathy Kessel, a
fellow researcher who did the heavy editing on my book, and I wrote a
student book and teacher's manual. The series is called the Knowing
Mathematics Transition Program. Our goal was to create a 12- to
14-week intervention that would bring the students up to their grade
level.
Many teachers mentioned that these students don't know the basic
facts, multiplication tables being their most obvious trouble. Some
teachers let the students use calculators, but using calculators alone
is not good. Children need to know multiplication tables on their own.
They can push buttons and find the numbers, but they don't know the
meaning of the numbers. They might not even know that 35 is two more
than 33. Numbers make no sense to them, so we decided to help them to
overcome this problem of not knowing basic facts.
We concentrated on the real math behind facts and then the facts
themselves. We use a question as the title for each daily lesson:
"What is a number?" or "Can you turn addition into subtraction?" or
"Do you know the trick for computing with 9?"
Each lesson includes two part--math conversation between the teacher
and a group of students, presented as a small play, and a set of math
problems. For the lesson on adding with 9, for example, we show this
pattern and ask the students to fill in the blanks.
9 + 1 = 10
9 + 2 = 11
9 + 3 = 12
9 + 4 = 13
9 + 5 = 14
9 + 6 = ___
9 + 7 = ___
9 + 8 = ___
9 + 9 = ___
After a few more patterns, we ask them the trick for computing with 9
and why it works. Mainly this lesson is for reflection on the concept.
The numbers are very simple, but we want the students to play out the
ideas. We think that the easier the numbers are, the easier it is for
the students to think about the concept behind them.
For every two-page lesson, one page is a set of problems; children
need practice and each problem is closely linked to the day's concept,
or to concepts and skills learned in earlier lessons.
The teacher's manual models the classroom discussion for students and
teachers, giving teachers questions to ask and explaining why the
questions are included. In field tests, both students and teachers
enjoyed the discussions. An example of a concept for discussion is the
real meaning of subtraction. We usually talk only about "take away,"
but then why do people call the result of subtraction "difference?" In
one lesson we discuss how "difference" is the one concept underlying
different models of subtraction. They come to realize that this
meaning is much more powerful than "What is left?"
I learned a lot of elementary math through designing this program,
through asking myself "Why." Why are addition and subtraction facts so
difficult to learn? One consideration is the number of these basic
facts: 81 for addition and 81 for subtraction.
The 81 Addition Pairs
What is the real idea behind these pairs? In this table, 20 pairs of
numbers have sums less than 10, and most intermediate grade students
feel comfortable with these addition facts. There are five pairs that
have a sum of 10:
9,1; 8,2; 7,3; 6,4; 5,5
Then there are another 20 pairs with sums beyond 10:
9,2; 9,3; 9,4; 9,5; 9,6; 9,7; 9,8; 9,9
8,3; 8,4; 8,5; 8,6; 8,7; 8,8;
7,4; 7,5; 7,6; 7,7;
6,5; 6,6
Actually, most fourth graders who are behind in addition and
subtraction are behind only because they are not good at these 20
pairs.
Now we can concentrate on those pairs, including the eight pairs
related to the number 9, which explains why we included the lesson on
finding a trick to compute with 9. When students have learned the
trick and practiced it, they have learned those eight pairs and can
move on to the doubles. They reduce those doubles little by little,
ending up with less than ten pairs that have to be learned by rote.
Through this approach we teach students basic facts while they learn
more about numbers. They cannot learn this by punching the buttons on
calculators.
When teachers use this kind of book, where they have to teach math in
a different way, we hope they will learn more math naturally through
teaching it. They will also learn more about what makes elementary
math challenging for students.
What would you say to elementary teachers who want to improve their
mathematical knowledge and their teaching?
If I were a teacher, I would be very aware that there is no existing
body of knowledge that I could study chapter by chapter. If I want to
improve myself, I must first depend on myself. I would ask myself
"Why?" all the time as I teach elementary math. I would also discuss
math teaching and learning with other teachers in my school who teach
at the same grade level. Conversation stimulates our thinking.
A layperson only needs to know how to do math, but a teacher needs to
learn more because his or her students need a whole picture--a whole
concept. Students develop the power of their minds through learning
conceptually. Mathematicians may not need these conceptual tools
because they have them internalized, but students need tools to
develop their abstractive way of thinking.
Teachers who feel that they need more content math knowledge should
depend on themselves, never forgetting that they are their own
teachers. Their own classroom teaching is the main process through
which they can learn. That really makes sense to me.
Could I ask you for any final thoughts?
Society is telling teachers so many things, and everybody who tells
them something seems to have authority -- psychologists, researchers.
Today's workshop says this is good, and tomorrow's workshop tells them
something else. Who is the teacher to believe?
Teachers should have confidence in themselves. They are the ones who
digest all the knowledge, who listen and change it into a "whole"
through their own understanding. I don't know if the word soul is the
right word here, but I think teachers are the soul that gives out the
light of all the knowledge they hold. I really want them, and us as
math educators, to think about that. Otherwise, teachers feel that
everybody knows more than they do. Then they lose themselves. I would
tell teachers: "Do not forget yourself as a teacher of yourself."
ENC is hosting an online discussion of this article. Join other
educators talking about questions such as these:
How can you use Liping Ma's ideas in your teaching?
Can you describe a situation in which you used your own class room
teaching as a process to help you learn?
Terese Herrera is the mathematics resource specialist at ENC. Her
career includes 15 years of teaching mathematics at the middle school
and high school levels. Email: therrera@enc.org
Reference
Ma, Liping & Kessel, Cathy. (2001). Knowing Mathematics Series.
Boston: Houghton Mifflin.
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