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Jerry P. Becker

Posts: 16,576
Registered: 12/3/04
Why Do Americans Stink at Math?
Posted: Jul 24, 2014 4:08 PM
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From The New York Times / Magazine, Wednesday, July 23, 2014. See

Why Do Americans Stink at Math?

By Elizabeth Green

When Akihiko Takahashi was a junior in college in
1978, he was like most of the other students at
his university in suburban Tokyo. He had a vague
sense of wanting to accomplish something but no
clue what that something should be. But that
spring he met a man who would become his mentor,
and this relationship set the course of his
entire career.

Takeshi Matsuyama was an elementary-school
teacher, but like a small number of instructors
in Japan, he taught not just young children but
also college students who wanted to become
teachers. At the university-affiliated elementary
school where Matsuyama taught, he turned his
classroom into a kind of laboratory, concocting
and trying out new teaching ideas. When Takahashi
met him, Matsuyama was in the middle of his
boldest experiment yet - revolutionizing the way
students learned math by radically changing the
way teachers taught it.

Instead of having students memorize and then
practice endless lists of equations - which
Takahashi remembered from his own days in school
- Matsuyama taught his college students to
encourage passionate discussions among children
so they would come to uncover math's procedures,
properties and proofs for themselves. One day,
for example, the young students would derive the
formula for finding the area of a rectangle; the
next, they would use what they learned to do the
same for parallelograms. Taught this new way,
math itself seemed transformed. It was not dull
misery but challenging, stimulating and even fun.

Takahashi quickly became a convert. He discovered
that these ideas came from reformers in the
United States, and he dedicated himself to
learning to teach like an American. Over the next
12 years, as the Japanese educational system
embraced this more vibrant approach to math,
Takahashi taught first through sixth grade.
Teaching, and thinking about teaching, was
practically all he did. A quiet man with calm,
smiling eyes, his passion for a new kind of math
instruction could take his colleagues by
surprise. "He looks very gentle and kind,"
Kazuyuki Shirai, a fellow math teacher, told me
through a translator. "But when he starts talking
about math, everything changes."

Takahashi was especially enthralled with an
American group called the National Council of
Teachers of Mathematics, or N.C.T.M., which
published manifestoes throughout the 1980s,
prescribing radical changes in the teaching of
math. Spending late nights at school, Takahashi
read every one. Like many professionals in Japan,
teachers often said they did their work in the
name of their mentor. It was as if Takahashi bore
two influences: Matsuyama and the American

Takahashi, who is 58, became one of his country's
leading math teachers, once attracting 1,000
observers to a public lesson. He participated in
a classroom equivalent of "Iron Chef," the
popular Japanese television show. But in 1991,
when he got the opportunity to take a new job in
America, teaching at a school run by the Japanese
Education Ministry for expats in Chicago, he did
not hesitate. With his wife, a graphic designer,
he left his friends, family, colleagues -
everything he knew - and moved to the United
States, eager to be at the center of the new math.

As soon as he arrived, he started spending his
days off visiting American schools. One of the
first math classes he observed gave him such a
jolt that he assumed there must have been some
kind of mistake. The class looked exactly like
his own memories of school. "I thought, Well,
that's only this class," Takahashi said. But the
next class looked like the first, and so did the
next and the one after that. The Americans might
have invented the world's best methods for
teaching math to children, but it was difficult
to find anyone actually using them.

It wasn't the first time that Americans had
dreamed up a better way to teach math and then
failed to implement it. The same pattern played
out in the 1960s, when schools gripped by a
post-Sputnik inferiority complex unveiled an
ambitious "new math," only to find, a few years
later, that nothing actually changed. In fact,
efforts to introduce a better way of teaching
math stretch back to the 1800s. The story is the
same every time: a big, excited push, followed by
mass confusion and then a return to conventional

The trouble always starts when teachers are told
to put innovative ideas into practice without
much guidance on how to do it. In the hands of
unprepared teachers, the reforms turn to
nonsense, perplexing students more than helping
them. One 1965 Peanuts cartoon depicts the young
blond-haired Sally struggling to understand her
new-math assignment: "Sets . . . one to one
matching . . . equivalent sets . . . sets of one
. . . sets of two . . . renaming two. . . ."
After persisting for three valiant frames, she
throws back her head and bursts into tears: "All
I want to know is, how much is two and two?"

Today the frustrating descent from good
intentions to tears is playing out once again, as
states across the country carry out the latest
wave of math reforms: the Common Core. A new set
of academic standards developed to replace
states' individually designed learning goals, the
Common Core math standards are like earlier math
reforms, only further refined and more ambitious.
Whereas previous movements found teachers
haphazardly, through organizations like
Takahashi's beloved N.C.T.M. math-teacher group,
the Common Core has a broader reach. A group of
governors and education chiefs from 48 states
initiated the writing of the standards, for both
math and language arts, in 2009. The same year,
the Obama administration encouraged the idea,
making the adoption of rigorous "common
standards" a criterion for receiving a portion of
the more than $4 billion in Race to the Top
grants. Forty-three states have adopted the

The opportunity to change the way math is taught,
as N.C.T.M. declared in its endorsement of the
Common Core standards, is "unprecedented." And
yet, once again, the reforms have arrived without
any good system for helping teachers learn to
teach them. Responding to a recent survey by
Education Week, teachers said they had typically
spent fewer than four days in Common Core
training, and that included training for the
language-arts standards as well as the math.

Carefully taught, the assignments can help make
math more concrete. Students don't just memorize
their times tables and addition facts but also
understand how arithmetic works and how to apply
it to real-life situations. But in practice, most
teachers are unprepared and children are baffled,
leaving parents furious. The comedian Louis C.K.
parodied his daughters' homework in an appearance
on "The Late Show With David Letterman": "It's
like, Bill has three goldfish. He buys two more.
How many dogs live in London?"

The inadequate implementation can make math
reforms seem like the most absurd form of policy
change - one that creates a whole new problem to
solve. Why try something we've failed at a
half-dozen times before, only to watch it
backfire? Just four years after the standards
were first released, this argument has gained
traction on both sides of the aisle. Since March,
four Republican governors have opposed the
standards. In New York, a Republican candidate is
trying to establish another ballot line, called
Stop Common Core, for the November gubernatorial
election. On the left, meanwhile, teachers'
unions in Chicago and New York have opposed the

The fact that countries like Japan have
implemented a similar approach with great success
offers little consolation when the results here
seem so dreadful. Americans might have written
the new math, but maybe we simply aren't suited
to it. "By God," wrote Erick Erickson, editor of
the website RedState, in an anti-Common Core
attack, is it such "a horrific idea that we might
teach math the way math has always been taught."

The new math of the '60s, the new new math of the
'80s and today's Common Core math all stem from
the idea that the traditional way of teaching
math simply does not work. As a nation, we suffer
from an ailment that John Allen Paulos, a Temple
University math professor and an author, calls
innumeracy - the mathematical equivalent of not
being able to read. On national tests, nearly
two-thirds of fourth graders and eighth graders
are not proficient in math. More than half of
fourth graders taking the 2013 National
Assessment of Educational Progress could not
accurately read the temperature on a neatly drawn
thermometer. (They did not understand that each
hash mark represented two degrees rather than
one, leading many students to mistake 46 degrees
for 43 degrees.) On the same multiple-choice
test, three-quarters of fourth graders could not
translate a simple word problem about a girl who
sold 15 cups of lemonade on Saturday and twice as
many on Sunday into the expression "15 + (2?15)."
Even in Massachusetts, one of the country's
highest-performing states, math students are more
than two years behind their counterparts in

The new math of the '60s, the new, new math of
the '80s and today's Common Core math all stem
from the idea that the traditional way of
teaching math simply does not work.

Adulthood does not alleviate our quantitative
deficiency. A 2012 study comparing
16-to-65-year-olds in 20 countries found that
Americans rank in the bottom five in numeracy. On
a scale of 1 to 5, 29 percent of them scored at
Level 1 or below, meaning they could do basic
arithmetic but not computations requiring two or
more steps. One study that examined medical
prescriptions gone awry found that 17 percent of
errors were caused by math mistakes on the part
of doctors or pharmacists. A survey found that
three-quarters of doctors inaccurately estimated
the rates of death and major complications
associated with common medical procedures, even
in their own specialty areas.

One of the most vivid arithmetic failings
displayed by Americans occurred in the early
1980s, when the A&W restaurant chain released a
new hamburger to rival the McDonald's Quarter
Pounder. With a third-pound of beef, the A&W
burger had more meat than the Quarter Pounder; in
taste tests, customers preferred A&W's burger.
And it was less expensive. A lavish A&W
television and radio marketing campaign cited
these benefits. Yet instead of leaping at the
great value, customers snubbed it.

Only when the company held customer focus groups
did it become clear why. The Third Pounder
presented the American public with a test in
fractions. And we failed. Misunderstanding the
value of one-third, customers believed they were
being overcharged. Why, they asked the
researchers, should they pay the same amount for
a third of a pound of meat as they did for a
quarter-pound of meat at McDonald's. The "4" in
"?," larger than the "3" in "?," led them astray.

But our innumeracy isn't inevitable. In the 1970s
and the 1980s, cognitive scientists studied a
population known as the unschooled, people with
little or no formal education. Observing workers
at a Baltimore dairy factory in the '80s, the
psychologist Sylvia Scribner noted that even
basic tasks required an extensive amount of math.
For instance, many of the workers charged with
loading quarts and gallons of milk into crates
had no more than a sixth-grade education. But
they were able to do math, in order to assemble
their loads efficiently, that was "equivalent to
shifting between different base systems of
numbers." Throughout these mental calculations,
errors were "virtually nonexistent." And yet when
these workers were out sick and the dairy's
better-educated office workers filled in for
them, productivity declined.

The unschooled may have been more capable of
complex math than people who were specifically
taught it, but in the context of school, they
were stymied by math they already knew. Studies
of children in Brazil, who helped support their
families by roaming the streets selling roasted
peanuts and coconuts, showed that the children
routinely solved complex problems in their heads
to calculate a bill or make change. When
cognitive scientists presented the children with
the very same problem, however, this time with
pen and paper, they stumbled. A 12-year-old boy
who accurately computed the price of four
coconuts at 35 cruzeiros each was later given the
problem on paper.

Incorrectly using the multiplication method he
was taught in school, he came up with the wrong
answer. Similarly, when Scribner gave her dairy
workers tests using the language of math class,
their scores averaged around 64 percent. The
cognitive-science research suggested a startling
cause of Americans' innumeracy: school.

Most American math classes follow the same
pattern, a ritualistic series of steps so
ingrained that one researcher termed it a
cultural script. Some teachers call the pattern
"I, We, You." After checking homework, teachers
announce the day's topic, demonstrating a new
procedure: "Today, I'm going to show you how to
divide a three-digit number by a two-digit
number" (I). Then they lead the class in trying
out a sample problem: "Let's try out the steps
for 242 ÷ 16" (We). Finally they let students
work through similar problems on their own,
usually by silently making their way through a
work sheet: "Keep your eyes on your own paper!"

By focusing only on procedures - "Draw a division
house, put '242' on the inside and '16' on the
outside, etc." - and not on what the procedures
mean, "I, We, You" turns school math into a sort
of arbitrary process wholly divorced from the
real world of numbers. Students learn not math
but, in the words of one math educator,
answer-getting. Instead of trying to convey, say,
the essence of what it means to subtract
fractions, teachers tell students to draw
butterflies and multiply along the diagonal
wings, add the antennas and finally reduce and
simplify as needed. The answer-getting strategies
may serve them well for a class period of
practice problems, but after a week, they forget.
And students often can't figure out how to apply
the strategy for a particular problem to new

How could you teach math in school that mirrors
the way children learn it in the world? That was
the challenge Magdalene Lampert set for herself
in the 1980s, when she began teaching
elementary-school math in Cambridge, Mass. She
grew up in Trenton, accompanying her father on
his milk deliveries around town, solving the
milk-related math problems he encountered. "Like,
you know: If Mrs. Jones wants three quarts of
this and Mrs. Smith, who lives next door, wants
eight quarts, how many cases do you have to put
on the truck?" Lampert, who is 67 years old,
explained to me.

She knew there must be a way to tap into what
students already understood and then build on it.
In her classroom, she replaced "I, We, You" with
a structure you might call "You, Y'all, We."
Rather than starting each lesson by introducing
the main idea to be learned that day, she
assigned a single "problem of the day," designed
to let students struggle toward it - first on
their own (You), then in peer groups (Y'all) and
finally as a whole class (We). The result was a
process that replaced answer-getting with what
Lampert called sense-making. By pushing students
to talk about math, she invited them to share the
misunderstandings most American students keep
quiet until the test. In the process, she gave
them an opportunity to realize, on their own, why
their answers were wrong.

Lampert, who until recently was a professor of
education at the University of Michigan in Ann
Arbor, now works for the Boston Teacher
Residency, a program serving Boston public
schools, and the New Visions for Public Schools
network in New York City, instructing educators
on how to train teachers. In her book, "Teaching
Problems and the Problems of Teaching," Lampert
tells the story of how one of her fifth-grade
classes learned fractions. One day, a student
made a "conjecture" that reflected a common
misconception among children. The fraction 5 / 6,
the student argued, goes on the same place on the
number line as 5 / 12. For the rest of the class
period, the student listened as a lineup of peers
detailed all the reasons the two numbers couldn't
possibly be equivalent, even though they had the
same numerator. A few days later, when Lampert
gave a quiz on the topic ("Prove that 3 / 12 = 1
/ 4 ," for example), the student could
confidently declare why: "Three sections of the
12 go into each fourth."

Over the years, observers who have studied
Lampert's classroom have found that students
learn an unusual amount of math. Rather than
forgetting algorithms, they retain and even
understand them. One boy who began fifth grade
declaring math to be his worst subject ended it
able to solve multiplication, long division and
fraction problems, not to mention simple
multivariable equations. It's hard to look at
Lampert's results without concluding that with
the help of a great teacher, even Americans can
become the so-called math people we don't think
we are.

Among math reformers, Lampert's work gained
attention. Her research was cited in the same
N.C.T.M. standards documents that Takahashi later
pored over. She was featured in Time magazine in
1989 and was retained by the producers of "Sesame
Street" to help create the show "Square One
Television," aimed at making math accessible to
children. Yet as her ideas took off, she began to
see a problem. In Japan, she was influencing
teachers she had never met, by way of the
N.C.T.M. standards. But where she lived, in
America, teachers had few opportunities for
learning the methods she developed.

American institutions charged with training
teachers in new approaches to math have proved
largely unable to do it. At most education
schools, the professors with the research budgets
and deanships have little interest in the science
of teaching. Indeed, when Lampert attended
Harvard's Graduate School of Education in the
1970s, she could find only one listing in the
entire course catalog that used the word
"teaching" in its title. (Today only 19 out of
231 courses include it.) Methods courses,
meanwhile, are usually taught by the lowest ranks
of professors - chronically underpaid, overworked
and, ultimately, ineffective.

Without the right training, most teachers do not
understand math well enough to teach it the way
Lampert does. "Remember," Lampert says, "American
teachers are only a subset of Americans." As
graduates of American schools, they are no more
likely to display numeracy than the rest of us.
"I'm just not a math person," Lampert says her
education students would say with an apologetic

Consequently, the most powerful influence on
teachers is the one most beyond our control. The
sociologist Dan Lortie calls the phenomenon the
apprenticeship of observation. Teachers learn to
teach primarily by recalling their memories of
having been taught, an average of 13,000 hours of
instruction over a typical childhood. The
apprenticeship of observation exacerbates what
the education scholar Suzanne Wilson calls
education reform's double bind. The very people
who embody the problem - teachers - are also the
ones charged with solving it.

Lampert witnessed the effects of the double bind
in 1986, a year after California announced its
intention to adopt "teaching for understanding,"
a style of math instruction similar to Lampert's.
A team of researchers that included Lampert's
husband, David Cohen, traveled to California to
see how the teachers were doing as they began to
put the reforms into practice. But after studying
three dozen classrooms over four years, they
found the new teaching simply wasn't happening.
Some of the failure could be explained by active
resistance. One teacher deliberately replaced a
new textbook's problem-solving pages with the old
worksheets he was accustomed to using.
Continue reading the main story

Teachers primarily learn to teach by recalling
their memories of having been taught, about
13,000 hours of instruction during a typical
childhood - a problem since their instruction
wasn't very good.

Much more common, though, were teachers who
wanted to change, and were willing to work hard
to do it, but didn't know how. Cohen observed one
teacher, for example, who claimed to have incited
a "revolution" in her classroom. But on closer
inspection, her classroom had changed but not in
the way California reformers intended it to.
Instead of focusing on mathematical ideas, she
inserted new activities into the traditional "I,
We You" framework. The supposedly cooperative
learning groups she used to replace her rows of
desks, for example, seemed in practice less a
tool to encourage discussion than a means to
dismiss the class for lunch (this group can line
up first, now that group, etc.).

And how could she have known to do anything
different? Her principal praised her efforts,
holding them up as an example for others.
Official math-reform training did not help,
either. Sometimes trainers offered patently bad
information - failing to clarify, for example,
that even though teachers were to elicit wrong
answers from students, they still needed,
eventually, to get to correct ones. Textbooks,
too, barely changed, despite publishers' claims
to the contrary.

With the Common Core, teachers are once more
being asked to unlearn an old approach and learn
an entirely new one, essentially on their own.
Training is still weak and infrequent, and
principals - who are no more skilled at math than
their teachers - remain unprepared to offer
support. Textbooks, once again, have received
only surface adjustments, despite the shiny
Common Core labels that decorate their covers.
"To have a vendor say their product is Common
Core is close to meaningless," says Phil Daro, an
author of the math standards.

Left to their own devices, teachers are once
again trying to incorporate new ideas into old
scripts, often botching them in the process. One
especially nonsensical result stems from the
Common Core's suggestion that students not just
find answers but also "illustrate and explain the
calculation by using equations, rectangular
arrays, and/or area models." The idea of
utilizing arrays of dots makes sense in the hands
of a skilled teacher, who can use them to help a
student understand how multiplication actually
works. For example, a teacher trying to explain
multiplication might ask a student to first draw
three rows of dots with two dots in each row and
then imagine what the picture would look like
with three or four or five dots in each row.
Guiding the student through the exercise, the
teacher could help her see that each march up the
times table (3x2, 3x3, 3x4) just means adding
another dot per row. But if a teacher doesn't use
the dots to illustrate bigger ideas, they become
just another meaningless exercise. Instead of
memorizing familiar steps, students now practice
even stranger rituals, like drawing dots only to
count them or breaking simple addition problems
into complicated forms (62+26, for example, must
become 60+2+20+6) without understanding why. This
can make for even poorer math students. "In the
hands of unprepared teachers," Lampert says,
"alternative algorithms are worse than just
teaching them standard algorithms."

No wonder parents and some mathematicians
denigrate the reforms as "fuzzy math." In the
warped way untrained teachers interpret them,
they are fuzzy.

When Akihiko Takahashi arrived in America, he was
surprised to find how rarely teachers discussed
their teaching methods. A year after he got to
Chicago, he went to a one-day conference of
teachers and mathematicians and was perplexed by
the fact that the gathering occurred only twice a
year. In Japan, meetings between math-education
professors and teachers happened as a matter of
course, even before the new American ideas
arrived. More distressing to Takahashi was that
American teachers had almost no opportunities to
watch one another teach.

In Japan, teachers had always depended on
jugyokenkyu, which translates literally as
"lesson study," a set of practices that Japanese
teachers use to hone their craft. A teacher first
plans lessons, then teaches in front of an
audience of students and other teachers along
with at least one university observer. Then the
observers talk with the teacher about what has
just taken place. Each public lesson poses a
hypothesis, a new idea about how to help children
learn. And each discussion offers a chance to
determine whether it worked. Without jugyokenkyu,
it was no wonder the American teachers' work fell
short of the model set by their best thinkers.
Without jugyokenyku, Takahashi never would have
learned to teach at all. Neither, certainly,
would the rest of Japan's teachers.

The best discussions were the most microscopic,
minute-by-minute recollections of what had
occurred, with commentary. If the students were
struggling to represent their subtractions
visually, why not help them by, say, arranging
tile blocks in groups of 10, a teacher would
suggest. Or after a geometry lesson, someone
might note the inherent challenge for children in
seeing angles as not just corners of a triangle
but as quantities - a more difficult stretch than
making the same mental step for area. By the end,
the teachers had learned not just how to teach
the material from that day but also about math
and the shape of students' thoughts and how to
mold them.

If teachers weren't able to observe the methods
firsthand, they could find textbooks, written by
the leading instructors and focusing on the idea
of allowing students to work on a single problem
each day. Lesson study helped the textbook
writers home in on the most productive problems.
For example, if you are trying to decide on the
best problem to teach children to subtract a
one-digit number from a two-digit number using
borrowing, or regrouping, you have many choices:
11 minus 2, 18 minus 9, etc. Yet from all these
options, five of the six textbook companies in
Japan converged on the same exact problem,
Toshiakira Fujii, a professor of math education
at Tokyo Gakugei University, told me. They
determined that 13 minus 9 was the best. Other
problems, it turned out, were likely to lead
students to discover only one solution method.
With 12 minus 3, for instance, the natural
approach for most students was to take away 2 and
then 1 (the subtraction-subtraction method). Very
few would take 3 from 10 and then add back 2 (the
subtraction-addition method).

But Japanese teachers knew that students were
best served by understanding both methods. They
used 13 minus 9 because, faced with that
particular problem, students were equally likely
to employ subtraction-subtraction (take away 3 to
get 10, and then subtract the remaining 6 to get
4) as they were to use subtraction-addition
(break 13 into 10 and 3, and then take 9 from 10
and add the remaining 1 and 3 to get 4). A
teacher leading the "We" part of the lesson, when
students shared their strategies, could do so
with full confidence that both methods would

By 1995, when American researchers videotaped
eighth-grade classrooms in the United States and
Japan, Japanese schools had overwhelmingly traded
the old "I, We, You" script for "You, Y'all, We."
(American schools, meanwhile didn't look much
different than they did before the reforms.)
Japanese students had changed too. Participating
in class, they spoke more often than Americans
and had more to say. In fact, when Takahashi came
to Chicago initially, the first thing he noticed
was how uncomfortably silent all the classrooms
were. One teacher must have said, "Shh!" a
hundred times, he said.

Later, when he took American visitors on tours of
Japanese schools, he had to warn them about the
noise from children talking, arguing, shrieking
about the best way to solve problems. The
research showed that Japanese students initiated
the method for solving a problem in 40 percent of
the lessons; Americans initiated 9 percent of the
time. Similarly, 96 percent of American students'
work fell into the category of "practice," while
Japanese students spent only 41 percent of their
time practicing. Almost half of Japanese
students' time was spent doing work that the
researchers termed "invent/think." (American
students spent less than 1 percent of their time
on it.) Even the equipment in classrooms
reflected the focus on getting students to think.
Whereas American teachers all used overhead
projectors, allowing them to focus students'
attention on the teacher's rules and equations,
rather than their own, in Japan, the preferred
device was a blackboard, allowing students to
track the evolution of everyone's ideas.

Japanese schools are far from perfect. Though
lesson study is pervasive in elementary and
middle school, it is less so in high school,
where the emphasis is on cramming for college
entrance exams. As is true in the United States,
lower-income students in Japan have recently been
falling behind their peers, and people there
worry about staying competitive on international
tests. Yet while the United States regularly
hovers in the middle of the pack or below on
these tests, Japan scores at the top. And other
countries now inching ahead of Japan imitate the
jugyokenkyu approach. Some, like China, do this
by drawing on their own native jugyokenkyu-style
traditions (zuanyan jiaocai, or "studying
teaching materials intensively," Chinese teachers
call it). Others, including Singapore, adopt
lesson study as a deliberate matter of government
policy. Finland, meanwhile, made the shift by
carving out time for teachers to spend learning.
There, as in Japan, teachers teach for 600 or
fewer hours each school year, leaving them ample
time to prepare, revise and learn. By contrast,
American teachers spend nearly 1,100 hours with
little feedback.

It could be tempting to dismiss Japan's success
as a cultural novelty, an unreproducible result
of an affluent, homogeneous, and math-positive
society. Perhaps the Japanese are simply the
"math people" Americans aren't. Yet when I
visited Japan, every teacher I spoke to told me a
story that sounded distinctly American. "I used
to hate math," an elementary-school teacher named
Shinichiro Kurita said through a translator. "I
couldn't calculate. I was slow. I was always at
the bottom of the ladder, wondering why I had to
memorize these equations." Like Takahashi, when
he went to college and saw his instructors
teaching differently, "it was an enlightenment."

Learning to teach the new way himself was not
easy. "I had so much trouble," Kurita said. "I
had absolutely no idea how to do it." He listened
carefully for what Japanese teachers call
children's twitters - mumbled nuggets of inchoate
thoughts that teachers can mold into the fully
formed concept they are trying to teach. And he
worked hard on bansho, the term Japanese teachers
use to describe the art of blackboard writing
that helps students visualize the flow of ideas
from problem to solution to broader mathematical
principles. But for all his efforts, he said,
"the children didn't twitter, and I couldn't
write on the blackboard." Yet Kurita didn't give
up - and he had resources to help him persevere.
He went to study sessions with other teachers,
watched as many public lessons as he could and
spent time with his old professors. Eventually,
as he learned more, his students started to do
the same. Today Kurita is the head of the math
department at Setagaya Elementary School in
Tokyo, the position once held by Takahashi's
mentor, Matsuyama.

Of all the lessons Japan has to offer the United
States, the most important might be the belief in
patience and the possibility of change. Japan,
after all, was able to shift a country full of
teachers to a new approach. Telling me his story,
Kurita quoted what he described as an old
Japanese saying about perseverance: "Sit on a
stone for three years to accomplish anything."

Admittedly, a tenacious commitment to improvement
seems to be part of the Japanese national
heritage, showing up among teachers, autoworkers,
sushi chefs and tea-ceremony masters. Yet for his
part, Akihiko Takahashi extends his optimism even
to a cause that can sometimes seem hopeless - the
United States. After the great disappointment of
moving here in 1991, he made a decision his
colleagues back in Japan thought was strange. He
decided to stay and try to help American teachers
embrace the innovative ideas that reformers like
Magdalene Lampert pioneered.

Today Takahashi lives in Chicago and holds a
full-time job in the education department at
DePaul University. (He also has a special
appointment at his alma mater in Japan, where he
and his wife frequently visit.) When it comes to
transforming teaching in America, Takahashi sees
promise in individual American schools that have
decided to embrace lesson study. Some do this
deliberately, working with Takahashi to transform
the way they teach math. Others have built
versions of lesson study without using that name.

Sometimes these efforts turn out to be duds. When
carefully implemented, though, they show promise.
In one experiment in which more than 200 American
teachers took part in lesson study, student
achievement rose, as did teachers' math knowledge
- two rare accomplishments.

Training teachers in a new way of thinking will
take time, and American parents will need to be
patient. In Japan, the transition did not happen
overnight. When Takahashi began teaching in the
new style, parents initially complained about the
young instructor experimenting on their children.
But his early explorations were confined to just
a few lessons, giving him a chance to learn what
he was doing and to bring the parents along too.
He began sending home a monthly newsletter
summarizing what the students had done in class
and why. By his third year, he was sending out
the newsletter every day. If they were going to
support their children, and support Takahashi,
the parents needed to know the new math as well.
And over time, they learned.

To cure our innumeracy, we will have to accept
that the traditional approach we take to teaching
math - the one that can be mind-numbing, but also
comfortingly familiar - does not work. We will
have to come to see math not as a list of rules
to be memorized but as a way of looking at the
world that really makes sense.

The other shift Americans will have to make
extends beyond just math. Across all school
subjects, teachers receive a pale imitation of
the preparation, support and tools they need. And
across all subjects, the neglect shows in
students' work. In addition to misunderstanding
math, American students also, on average, write
weakly, read poorly, think unscientifically and
grasp history only superficially. Examining
nearly 3,000 teachers in six school districts,
the Bill & Melinda Gates Foundation recently
found that nearly two-thirds scored less than
"proficient" in the areas of "intellectual
challenge" and "classroom discourse."
Odds-defying individual teachers can be found in
every state, but the overall picture is of a
profession struggling to make the best of an
impossible hand.

Most policies aimed at improving teaching
conceive of the job not as a craft that needs to
be taught but as a natural-born talent that
teachers either decide to muster or don't
possess. Instead of acknowledging that changes
like the new math are something teachers must
learn over time, we mandate them as "standards"
that teachers are expected to simply "adopt." We
shouldn't be surprised, then, that their students
don't improve.

Here, too, the Japanese experience is telling.
The teachers I met in Tokyo had changed not just
their ideas about math; they also changed their
whole conception of what it means to be a
teacher. "The term 'teaching' came to mean
something totally different to me," a teacher
named Hideto Hirayama told me through a
translator. It was more sophisticated, more
challenging - and more rewarding. "The moment
that a child changes, the moment that he
understands something, is amazing, and this
transition happens right before your eyes," he
said. "It seems like my heart stops every day."
B. Myers. Prop stylIst: Randi Brookman HarrIs.
Calculator icons by Tim Boelaars.
by Andrew B. Myers. Prop stylist: Randi Brookman
Andrew B. Myers. Prop stylist: Randi Brookman
Harris. Butterfly icon by Tim Boelaars.
Elizabeth Green, the chief executive of
Chalkbeat, is the author of "Building a Better
Teacher," to be published by W. W. Norton next
month, from which this article is adapted.
Editor: Ilena Silverman
A version of this article appears in print on
July 27, 2014, on page MM22 of the Sunday
Magazine with the headline: (New Math) - (New
Teaching) = Failure.

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