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A panel addressed the following questions related to the mathematical knowledge of
teachers:
· What are the mathematical resources that teachers need to teach well?
· How can teachers learn the mathematics they need to teach well?
The papers that follow are edited transcripts of their remarks.
Moderator: Deborah Loewenherg Ball, University of Michigan
Zalman ~ Usiskin, Professor, University of Chicago
Deborah Schifter, Senior Scientist, Education Development Center
Marco Ishigaki, Professor, Waseda University
Miho Ueno, Mathematics Teacher, Tokyo Gak~gei University Senior High School

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WHAT MATHEMATICS DO
TEACHERS NEED THAT THEY ARE
LIKELY NOT TO ENCOUNTER IN
THEIR MATHEMATICS COURSES?
Zalman P. Usiskin, University of
Chicago
It is a truism. A teacher of mathematics
should know a great deal of mathematics.
The higher the level taught, the more the
teacher needs to know. For a teacher of
high school mathematics, this means
knowing a good deal of number theory,
algebra, geometry, analysis, statistics,
computer science, mathematical model-
ing, and history of mathematics. This is
what we might view as the traditional
background of a teacher who is consid-
ered to be well prepared mathematically.
Even though it is good to take more
and more mathematics, there is a problem
that taking more mathematics creates.
Often the more mathematics courses a
prospective teacher takes the wider the
gap between the courses taken and the
courses the teacher will teach. The gap is
both in the mathematical content and the
ways that content is approached. An
entire body of mathematical knowledge is
ignored.
L~ ~ t.,'d'` ~
There is a substantial body of math-
ematics that arises from teaching situa-
tions in much the same way that statistics
arises from data and applied mathematics
arises from real situations, and that
deserves to be viewed as a branch of
mathematics in its own right. ~ call this
"teachers' mathematics." A project
currently underway entitle(1 "High School
Mathematics from an A(lvance(1 Stan(l-
point" is (leveloping a first course in
teachers' mathematics for high school
teachers, and second and third courses
are being planned. ~ will attempt to
describe the motivation and content of
these courses.
THE PROBLEM
Every teacher of mathematics nee(ls
1. to see alternate (definitions and their
consequences;
2. to know why concepts arose anti how
they have changed over time;
3. to know the wi(le range of applications
of the mathematical i(leas being
taught;
4. to discuss alternate ways of approach-
ing problems, inclu(ling ways with anti
without calculator anti computer
technology;

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5. to see how problems and proofs can
be extended and generalized; and
6. to realize how ideas studied in school
relate to ideas students may encounter
in later mathematics study.
The result of the lack of teaching these
ideas to prospective teachers is that
teachers are often no better prepared in
the content they will teach than when they
were students taking that content. For
instance, they may know no more about
logarithms or factoring trinomials or
congruent triangles or volumes of cones
than is found in a good high school text.
THREE KINDS OF MATHEMATICS FOR
TEACHERS
Three kinds of mathematics content are
particularly needed by teachers. Each
might be said to consitute a facet of
looking at school mathematics content at
a deeper level than is possible for high
school students.
One focus is on mathematics particu-
larly useful to high school teachers that
might not normally be encountered in the
standard courses taken by mathematics
majors. Box ~ contains an example.
There is an analogous theorem for
inequalities, which ~ do not have the time
You can add the same number to both sides of an equation, or multiply both sides of
an equation by the same nonzero number, and the resulting equation is equivalent to the
given one. But if you square both sides of an equation, you may gain solutions. And
cubing both sides of an equation Joes not affect the solutions. What about taking the
log of both sinless Or taking the sine of both sinless How can one tell, in general,
whether an operation on both sides of an equation will change the solutions to the
equations
Here we are concerned with real-number solutions. Then an equation in one variable
can be thought of being of the form fix) = Ax), where x is real.
Applying an operation to both sides is like applying a function h to both sides. This
results in the equation
htf~x)) = h~gfx)~.
There is a very nice theorem: The two equations fix) = gtx) and htf~x)) = h~gtx)) are
equivalent if and only if h is a one-to-one function on the ranges of fix) and Ax).
Examining this theorem and its special cases unifies the solving of equations and gives
the teacher new insight into the process of equation solving.
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to mention here. But the more important
point is that there is a lot of content of this
type: theorems that integrate content that
might be taught in different units or
different years; theorems that shed light
on formulas, figures, or functions; and
so on.
The second focus is on the extended
analysis of problems. Recall the four
problem-solving steps of Polya (19521:
understanding the problem, devising a
plan, carrying out the plan, and looking
back. Most analyses of problem solving
devote their time more to devising a plan
than any other step. This is important,
but it is also quite important to examine
the last step: looking back. This means
looking at a problem after it has been
solved and examining what has been
done. Will the method of solution work
for other problems?
Here is an example from Dick Stanley
of Berkeley (Box 2), who is one of the
main authors of the materials we are
. · ~
clevlslng.
The third type of mathematics is the
explication and examination of concepts
(Box 3~.
Recall the well-known problem in which a rectangular sheet of cardboard is folded
into a box by cutting out four congruent squares from each corner. What is the maxi-
mum volume of the box that can be created If the cardboard is 12" by 18" and each
square has side x, then the box has height x and length and width 12 - 2x and 18 - 2x.
So the problem is to maximize xt12 - 2x)18 - 2x) over the range of possible values of x.
The problem can be Jone these Jays by graphing the function fix) = xt12 - 2x)18 - 2x),
or it can be Jone with calculus, or in a numerical way by appropriate substitution. It
happens that the volume is maximized when x = 5 + ~17 . But that tells us very little-
it does not give us intuition into the problem. Why are there two solutions Are they
related in some ways If we leave the problem without examining such questions, then
we have gone no farther than the typical class.
We can gain more intuition by letting the length of the rectangular sheet be 1 (say
1 foot) and the other dimension be w. If w is near zero, then the rectangle is long and
thin. If w is near 1, then the rectangle is near a square. If w is large, then the rectangle
again becomes long and thin. In our example, since 18 is 1 .5 times 12, w is 1 .5. How
is the value that maximizes volume related to we The relationship turns out to be inter-
esting and gives insight into the problem that was not obvious.
MATHEMATICAL KNOWLEDGE OF TEACHERS PANEL

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Consider the idea of parallel lines. (1 ) Parallel lines are lines that are equidistant
from one another. In this conception, parallel lines are an instance of parallel curves.
This conception explains why train tracks are called parallel. (Tracks are parallel even
when they curve.) (2) Parallel lines are lines that do not intersect. This conception places
parallel lines as an example of disjoint sets. This is the usual definition of parallel lines.
(3) Parallel lines are lines that go in the same direction. A~aebraica~Y. this means lines
... .. . . . ... .. .
.
v ,,
wits the same slope ano so under thus conception unless an exception is mane a fine
is parallel to itself. Sameness of direction intuitively underlies why, when parallel lines
are cut by a transversal, corresponding angles have the same measure.
Virtually every mathematics concept and all the important ones can be examined
in a variety of ways. When we give a definition for an idea, we almost immediately put
blinders on the other ways of looking at the idea. By reexamining the variety of ways
from a broader perspective, we can appreciate why students may have difficulty con-
necting various aspects of the same idea.
SUMMARY
'~eacher's mathematics" is a field of
applied mathematics that deserves its own
place in the curriculum. There is a huge
amount of material that falls under this
heading. However, this material is usually
picked up by teachers only haphazardly
through occasional articles in journals, or
by attending conferences like this one, or
by reading through teachers' notes found
in their textbooks, or by examining
research in history and conceptual
foundations of school mathematics. This
mathematics is often not known to profes-
sionalmathematicians. It covers both
pure anti applie(1 mathematics, algorithms
and proof, concepts and representation.
Teachers' mathematics is not merely a
bunch of mathematical topics that might
be of interest to teachers but a coherent
held of stu(ly, (listinguishe(1 by its own
important i(leas: the phenomenology of
mathematical concepts, the exten(le(1
analyses of related problems, and the
connections and generalizations within
anti among the (1iverse branches of
mathematics. The importance of teachers'
mathematics thus goes well beyond the
nee(ls of teachers to inclu(le all those who
study the learning of mathematics and the
mathematics curriculum.
Deborah Schifter, Education
Development Center
When ~ spoke on Sunday, ~ mentioned
that one problem with the implementation
of the National Council of Teachers of
Mathematics Standards is that many
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teachers and professional developers have
emphasized new teaching strategies at the
expense of what the reforms were actually
about, i.e., mathematical understanding.
If this is indicative of a deep-seated
tendency in the United States to adopt
superficial strategies to get at deep
problems we must be aware of this in
the context of lesson study too. ~ am
concerned that people wail get very
enthusiastic about the strategy of lesson
study and lose the essence of what it is
about.
One question ~ have is whether there
are shared ways of thinking about math-
ematics, learning, teaching, and ciass-
rooms implicit in the practice of lesson
study in Japan, ways of thinking that
would need to be cultivated among
teachers in the United States to make
lesson study profitable. For example, when
we viewed the video of Mr. Nakano's
fourth-grade lesson, we watched one
student explain the reasoning behind his
incorrect answer reasoning that was
quite easy for us to follow but which
bypassed the mathematics of the problem.
In our discussion of his lesson, Mr. Nakano
commented that when students make an
error or have difficulty with an idea, as
this child did, this is when "the fun
begins." My interpretation of his remark
is that this is when his work begins the
teacher becoming aware of difficulties his
students are having, figuring out what it is
they do understand in relation to the
learning objectives he has for his students,
and then developing a path to reach these
objectives.
My question is, is this understanding
shared among Japanese teachers who
engage in lesson study? And since it is
not shared among teachers in the United
States, is lesson study an appropriate
context for developing it, or are there
other, more propitious settings in which
teachers might better develop this disposi-
tion toward their work? Similarly, do
Japanese teachers who engage in lesson
study share an understanding of the
mathematics of the curriculum they teach?
Given the mathematical needs of many
U.S. elementary teachers, is lesson study
the appropriate context to a(l(lress these?
There is evidence that many teachers,
anti Americans generally, lose touch with
their capacity to think mathematically as
early as in the primary gra(les. It's at this
point that they start to learn that math-
ematics is memorization, in the process
losing touch with their own powers of
reasoning about mathematics. Anti so
when we look at the work that teachers
need to do, we must keep this in mind,
un(lerstan(ling at the same time that this
(foes not reflect on teachers' intelligence
but is the result of their own schooling. As
we discuss their serious needs in math-
ematics, it is very important to maintain a
spirit of respect for the teachers who still
have so much to learn.
In order to convey some of the issues
raise(1 by elementary teachers' math-
ematical deficits, ~ will describe three
(lifferent reactions to one set of activities
frequently (lo.
In these activities, we look at some very
common strategies children devise for
solving multidigit calculations and then
ask the teachers to apply the chil(lren's
methods to other pairs of numbers. When
~ begin a course this way, ~ consistently
provoke several (lifferent reactions from
teachers. Some actually get quite agitate
anti argue that all the children (li(1 the
calculations the wrong way. Apparently,
these teachers believe there is only one
way to solve a given problem and that is to
apply the algorithm they were taught in
school. This points to a very important
learning need for teachers: They must
come to see that understanding the
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mathematics knowing operations and
calculations involves more than being
able to apply a single algorithm. This
must be one of the goals for these teachers,
and it is the work of their instructors to
help teachers recognize that there is a
larger world of mathematics they can
enter. A second common response is
one of relief: teachers recognizing that
the children's procedures are ones they
themselves have always employed. But
under the impression that there was
something wrong with their work, they
had always kept it secret and felt some-
what ashamed. So it comes as a relief to
have their own strategies for calculating
acknowledged as valid. However, having
engaged in such "freelance" mathematical
reasoning in secret, this capacity to think
on their own remained underdeveloped.
It is important now to encourage these
teachers to move forward, to develop their
powers of mathematical thought. In many
cases, they learn, to their surprise, that
they are strong mathematical thinkers. It
is worth adding that many such teachers,
once they discover that their ways of
thinking were mathematically valid, go
through a period of sadness or anger over
lost opportunities, over the many years
they could have been doing satisfying
mathematics hall they hall the right
encouragement.
A third common response is illustrated
by what happened the first time ~ did an
exercise like this with teachers. We had
been working for some time, when one of
the teachers blurted out, "I can follow the
student's procedure. ~ can apply the
method to a different set of numbers. But
don't understand why it works. This is
another meaningless algorithm to me."
An(1 many of the other teachers in the
class agree(1 with her. At that time, ~ was
quite surprised, but since then ~ have
come to understand what was going on
here. That is, again, the teachers have
learned the mathematics by rote. But
unlike those who react in the first way
(lescribe(l, these teachers (lo un(lerstan
that reasoning must play some role in
mathematics. However, they never
developed models or representations, no
sense of what the operations actually do,
to call upon to make sense of mathematical
procedures. To illustrate the kinds of
mo(lels or representations, the kind of
mathematical imagination, teachers need
to develop, consider Mr. Kurosawa's
students. They were working with a
sequence of images of (lots with an
accompanying story starting with one
virus, the viruses grow by adding four
each minute. Some students represented
the number of viruses after three minutes
as 4 x 3 + I, others as 3 x 4 + 1. To explain
these different arithmetic representations,
students groupe(1 the (lots in (1ifferent
ways (Figure 11.
This is precisely the sort of mathemati-
cal imagination teachers need to (levelop.
Given the mathematical nee(ls of so many
elementary teachers in the U.S., our first
priority must be to help them reconnect
with their own capacities for mathematical
thought, to help them (levelop meanings
for the symbols and objects of mathemati-
cal study. But how is this to be done? It
certainly isn't happening in most math-
ematics courses offered at colleges anti
universities. One possibility is to work
from records of practice, perhaps like
those we have seen today, that highlight
chil(lren'smathematicalthinking. Such
records, which reveal children's math-
ematical i(leas in process, coul(1 provi(le
access to those same ideas for teachers
who (li(1 not have opportunities to (1evelop
these ways of thinking when they, them-
selves, were children.
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FIGURE ~ Students show different ways to group the Jots.
Haruo Ishigaki, WasecIa University
What do teachers learn and how do
they learn about mathematics? When it
comes to that issue, then it is basically the
same as how students learn and what
students learn. Therefore, ~ would like to
share with you the following four points.
First of all, the knowledge that any
excellent teacher has is not a perfect one.
Teachers need to be aware that their
knowledge always has to be updated, and
it has to be reconstructed. They should
encourage students, but they should also
encourage themselves. Good teaching is
80 percent confidence and 20 percent
doubt. Teachers need to have both of
these. When ~ just entered the university,
~ listened to the lecture of a Nobel laureate
in quantum mechanics. And suddenly in
the middle of the lecture, he stopped and
started thinking. He had forgotten
absolute zero. Was it minus 273 degrees
or minus 237 degrees? He wasn't sure. If
he was a Nobel laureate and forgot, it is
okay to have doubt, ~ thought. That was a
very big motivation for my work.
Second, students say extraordinary
things and commit extraordinary mis-
takes. And they give explanations which
are not understandable to us. There are
many situations, however, when there is
some very valuable information in those
little things that students or pupils say.
slid not let a particular student take my
graduate course, so he went to another
university. After two years, he contacted
me. He was going to talk at an academic
meeting and wanted me to come and
listen to his presentation. ~ realize(1 that
had lost a very big treasure. ~ should
have listened to his mathematics. This is
true even when students are still children.
Anti third, in Japan the teachings of
Confucius were common in education.
One of his teachings is that you shoul(1
always correct your mistakes. So at an
appropriate place, you shoul(1 recognize
that you have already committed a mis-
take anti acknowle(lge it. Often if some-
bo(ly asks a student questions, the student
thinks he is being criticized. ~ listened to
one of the lectures by a famous mathema-
tician. He ma(le a mistake, anti the
au(lience began to mumble anti give him
a(lvice. But ~ was very much impresse(1
by the professor's attitude. He aske(1 the
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audience to wait for about ten minutes, as
~ remember. He went to the corner of the
blackboard and worked through what he
had explained. And then finally, he turned
back to the audience and said, "you are
right." And then he went on.
And finally, let me go back to the
mathematics, the topic of today. A math-
ematical concept must be understood in
context. While ~ was working as an editor
of the academic society of mathematics,
happened to encounter a contribution
about rounding off calculations. Take a
very easy example, 3.1 x 3.9. Let's make
it 3 x 4. And the answer should be 12.
The students, however, repeatedly made a
calculation of 3.1 x 3.9, and rounded the
results to get 12.
The author wrote about how to correct
the students' mistake. His intention was
to teach the students that if you use
substitutes which are very close to the
exact numbers, the result will be very
close to the real answer. In his explana-
tion, however, he said that this is math-
ematically wrong, but in order to be
efficient, using the substitutes is a good
way to do the problem. Many students
are good students and try to get to the
right solution, not the wrong solution.
understood that this is why the teacher
failed. ~ think, in this particular case, the
teacher did not understand the real
purpose of teaching this method to
students. The approach should have been
taught in a specific environment, where
the teacher un(lerstoo(1 the real aim anti
the real purpose of the strategy.
Let me talk a little about concept
building to conclude my presentation.
The priority in concept buil(ling is to first
define it as a collection of elements with
something analogous and relate it to the
external world. The contents, the com-
mon features inside, should follow the
(definition. That will come later. ~ think
we can learn something from this about
knowing mathematics for teaching.
Miho Ueno, Tokyo Gakugei
University
I'm from the Oizumi Campus of Tokyo
Gakugei University Senior High School.
The Tokyo Gakugei University has
teacher preparation and an education
faculty. ~ work for the senior high school
and teach at the university. That means
every year we receive the students from
the Tokyo Gakugei University teacher
training program anti for lesson stu(ly.
When students come to our school to
have lesson study, ~ have a chance to see
what level of knowledge they start out
with in teaching mathematics. During the
three weeks, ~ can also identify how they
have developed their skills and their
un(lerstan(ling of mathematics education.
Student teachers come to our school
anti try to remember what they (li(1 when
they were high school students. This is
their understanding of high school
classroom activities. For example, one
student sai(1 when he took mathematics
classes in the past, students were always
taking notes, and the teachers explained
and let the students solve the problems.
The next student said that when he was a
high school student, he un(lerstoo(1 that
mathematics was testing students' ability
to memorize, anti they learne(1 mathemat-
ics in that way. The thir(1 student sai(1 that
about 40 students were in the same class.
Mathematics class was always quiet.
Children use(1 formulas anti wrote them
down in notebooks.
These students were typical. Every
year when they conclu(le their three-week
teacher training course, ~ ask them how
their attitude about the classroom has
change(l. If students share the same
impression about what they did when they
were high school students, then they have
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a misunderstanding about mathematics
teaching. They only know how to help
students solve the problems as they are
told to do in their textbook.
Prior to the training period, student
teachers have certain anxieties about
what they are supposed to do. But they
are relatively confident about their abili-
ties to teach mathematics. They believe
that their ability is enough to attain the
level required by the guidebook for their
lessons. During the three-week period,
student teachers have to compile a lesson
plan prior to their classes. They have to
describe the purpose of the class, what
kind of teaching materials should be used,
what they are supposed to say to the
students, and what reactions they expect.
The students understand that the defini-
tion of the formula is not the purpose.
They recognize that they have to learn
how they can help students solve prob-
lems and understand the mathematics.
The student teachers realize that you
cannot only rely on textbooks, but rather
you have to convey the value and essence
of the mathematics with your own wording
and with your own understanding. You
have to be very careful about how you lay
out the mathematics on the blackboard.
The student teacher, at the beginning of
the training, knows that his role as a
teacher is to convey information to his
students. But gradually they begin to
realize that the class is the place where
students have to learn. You have to devise
teaching materials. You have to think
carefully how you can use the pace of the
class and how you can use the blackboard
to help students learn. Their view of
students changes while they teach during
the training period. They begin to learn
that you have to find out the value of the
teaching materials, and you have to go
into them deeper yourself so you can
leverage what is written in the materials.
This is a capability teachers are required
to have.
Teachers must understand the variable
aspect of mathematical concepts and how
deep the mathematical formulas are.
When they are trying to find out and
explain these (leep aspects of the math-
emetics, they always discover that what
they have as an analogy is not sufficient.
This applies not only to the student
teachers but also to experienced teachers.
Teachers have to use their existing
knowledge as a basis, but they also have
to keep in mind that they have to improve
their knowledge. As teachers try to
acquire new knowledge of mathematics,
they have to decide how to learn. They
also have to know how formulas anti
algorithms came into existence. For
example, you have to consider a particular
mathematical task under certain condi-
tions. But in other cases you have to
change the conditions to the same task, or
you may try to generalize the same task,
anti by (1oing so you will be able to see a
pattern. Acquiring this kind of compe-
tence should be done not only by the
student teachers but also experienced
teachers as well as students in class. You
have to encourage students to obtain and
acquire this kind of thinking. Unless
teachers have this competence, they won't
be able to teach that concept to the
students.
The basic attitude teachers should have
toward studies is to be modest and try to
learn as much as possible. Teachers are
researchers at the same time as they are
teachers, but they cannot stay only in a
very narrow scope of their research. If
teachers want to expand their scope of
knowle(lge, they have to cooperate with
their colleagues, and they have to enjoy
(liscussions with other mathematics
teachers. With this attitude they can
(leepen their knowle(lge of mathematics.
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