Nonpartisan
Education Review / Reviews, Volume 10, Number 1

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**Why
Do Americans Stink at Math? Some of the Answer.**

Wayne Bishop, PhD

Mathematics, California
State University LA

On
July 23, 2014, a lengthy and persuasive article appeared in the *New York Times Magazine* written by the
author as an introduction to her new book by the same eye-catching name:

**Why Do Americans
Stink at Math?**

Elizabeth Green

http://www.nytimes.com/2014/07/27/magazine/why-do-americans-stink-at-math.html

Ms.
GreenÕs basic premise is that somehow Japanese classrooms modified their style
from the purportedly traditional approach of rote response and memorization of
meaningless algorithms to the American style of teaching mathematics following
the National Council of Teachers of Mathematics (NCTM) * Standards*. Unfortunately, her article - and,
presumably, book - represent so much education industry misinformation that it
almost cries out for clarification/correction. Herein is an attempt toward that end. It is far from complete but I will try
to cover the high points. In
summary, Ms. Green has referenced much of the same source material as mentioned
herein but a deeper look at those sources leads to quite a different
interpretation of the past and, more importantly, implications for trying to
improve performance in mathematics in many American precollegiate
schools. For the record, there are
many American precollegiate schools with exemplary
mathematics performance.

For
clarification, Òthe *NCTM Standards*Ó
will (as is common) represent the NCTM philosophy since there is no single
document by that name. Officially,
the original was the *Curriculum and
Evaluation Standards for School Mathematics* in 1989 and rewritten as *Principles and Standards for School
Mathematics* in 2000 supplemented by the *Curriculum
Focal Points *in 2006. The *Focal Points* were in response to
widespread negative reaction to the nebulous nature of the misnamed *Standards* and added much-needed
specificity. That lack of
specificity was deliberate and the *Focal
Points *are seldom mentioned in *NCTM
Standards *discussions because the actual philosophy is pedagogical, not
grade-level - or even appropriately sequential - mathematics content oriented.

************************************************

ÒTakeshi 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 [Akihiko] 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.Ó

Early
in the article we already see the heart of the problem yet to be fully realized
in Japan but overwhelmingly dominant in the US and the best general answer to
the very title of the article: The
rise of professional education as a separate academic entity as opposed to a
strong interest of many bright people with general interests at the elementary
level and specialized interests in the upper grades. In the era of my motherÕs education, a
particularly strong elementary and high school academic experience followed by
one year of Ònormal schoolÓ prepared much more effective elementary school
teachers than those of our current teacher factories parroting the
myths-of-the-era. Whole Language
Learning (that means phonics-free reading ÒinstructionÓ) and Bilingual
Education (that means Spanish only at first and gradually transitioning to
English without ever getting there) are a couple of
such examples that have been thoroughly debunked but remain undead. I will concentrate here on math ed efforts with an even longer
history and just as undead.

In Ms GreenÕs article, note the absence of mathematics
expertise or contributions from math-based academic areas such as engineering
or the hard sciences. Even
representation from math-requiring so-called trades that need computationally
competent novices to train in their expertise is missing. No, it is nothing but Òreceived wisdomÓ,
more appropriate in religion than education.

************************************************

ÒInstead of having students memorize and then practice
endless lists of equations — which Takahashi remembered from his own days
in school...Ó

This
Òhaving students memorize and then practice endless lists of equationsÓ is one
of the most often propped up mathematics education ÒstrawmenÓ. That was not my education more than a
half-century ago in rural Iowa and it was not TakahashiÕs. Decades ago, 1954 FieldÕs Medalist
(think Nobel Prize for mathematics) Kunihiko Kodaira oversaw (wrote?) superb precollegiate
mathematics books for Japanese students that reflect none of that yet the same
indefensible practices get repeated as historical fact year-after-year,
decade-after decade. Regrettably,
it is used as justification for NOT developing a certain level of automaticity
and accuracy with elementary number facts very early in the school years
(little kids eat that stuff up and 5-10 minutes a day on a regular basis is
plenty to accomplish the goal).
Such automaticity is critical to nearly all of what follows in
mathematics. Are the standard
algorithms of arithmetic difficult?
They are not! Organized work
and accurate elementary number facts makes teaching, computing, and
understanding of them a natural transition. No big deal. The same is true for the arithmetic of
ordinary fractions. Explaining
ÒwhyÓ is a little harder but not that much and the real problems are lack of
competence with elementary number facts and pervasive lack of teacher
competence. If a student has to
spend too much time thinking about and too often getting the little
computations wrong, the big ones get branded as impossibly hard. If the teacher has problems with these
(too often the case), we have an entirely different problem that is almost
insurmountable in a Òself-containedÓ classroom.

Speaking
of Òthe standard algorithms of arithmeticÓ, Ms. GreenÕs article can be
interpreted in support of the widely discussed *Common Core State Standards in Mathematics* (hereafter *CCSS-M*). To avoid the controversy that the NCTM
addressed (and then ignored) with its *Focal
Points*, the authors of the *CCSS-M*
incorporated an interesting ploy that needs more exposure, a deliberate
bait-and-switch on Òthe standard algorithmsÓ of elementary arithmetic. Professional math educators have long
argued that there is no such thing (because of trivial differences that do
exist within the standard ones), and thatÕs what calculators are for anyway so
who cares? Such a cavalier attitude
often receives an appropriate reaction from concerned observers from parents to
university mathematics professors.
A familiar example of how different algorithms can be is (for the
initiated) the Lattice Method for multiplication that some math curricula
present and almost require. The *CCSS-M* writers Òdodged that bulletÓ by
requiring competence with the standard algorithms. Problem? They use words (always too many words
and insufficient numerical examples!) without specifying what Òthe standard
algorithmsÓ means. As it turns out,
the words mean nothing at all. A
careful, if somewhat cynical, reading of this explanatory paper by a couple of
well-known math ed experts informs us that all
algorithms based on place-value numeration (base 10 to most people) fall within
the purview of the definitive article ÒtheÓ provided plenty of place-value
ÒunderstandingÓ is developed with it.

Ò__Standard
Algorithms in the Common Core State Standards__Ó by Karen Fuson
and Sybilla Beckmann. http://www.mathedleadership.org/docs/resources/journals/NCSMJournal_ST_Algorithms_Fuson_Beckmann.pdf

************************************************

ÒMatsuyama taught his college students to encourage
passionate discussions among children so they would come to uncover math's
procedures, properties and proofs for themselves.Ó

Nice as such words sound, the idea is genuinely absurd. Mathematics developed incrementally over
millennia by geniuses, not by a group of ordinary students passionately
discussing the procedures, properties, and proofs of mathematics that they have
been able to discover independently.
Fibonacci is famous not only because of his familiar sequence but for
having brought Hindu-Arabic numeration (base 10 because we have 10 fingers) to
European businessmen with the ease of the standard computational
algorithms. Newton (the inventor of
calculus and so much else) famously said, ÒIf I have been able to see further,
it was only because I stood on the shoulders of giants.Ó Professional mathematics educators
continue to insist on ÒdiscoveryÓ, often called ÒconstructivistÓ, pedagogy that
forces students to invent their own algorithms rather than be taught the
standard ones. The
very antithesis of NewtonÕs quote.

************************************************

ÒOne 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.Ó

In
contrast to his former description, this is excellent teaching - let me
emphasize TEACHING - and there is nothing new about it; in fact, Òtime
immemorialÓ comes to mind. Clearly, this sequence is not the blind leading the
blind but a carefully structured setting where the students do recognize, ÒdiscoverÓ
if you wish, the formulas along with a solid understanding of why they
work. Continue a little longer and
students ÒdiscoverÓ why the formula for the area of a triangle is what it is. Moreover...

ÒTakahashi 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.Ó

This
happens to be good teaching no matter where itÕs from but it is not teaching
Òlike an AmericanÓ of our so-called reform math era (New-New Math, Rainforest
Math, fuzzy math, and lots of other names, unfortunately not of
4-letters). The totally misleading
part is the idea that this was something new in Japan! Associated with the 1995 *TIMSS* (*Third International Mathematics and Science Study*), UCLAÕs Jim
Stigler conducted a video series of math classes in several countries being
tested including the US and Japan.
Consequently, it has already been documented that that style of teaching
was the way of life in Japanese schools 20 years ago. The tapes are a delight to watch; an
extension, for example, of a lesson that had been pursued the day before where,
if the students Òsaw itÓ, there was an almost trivial solution to the new
setting but, if not, there were more crude and laborious solutions
available. The instructor would
give a limited time, 10 minutes maybe, for students to explore their solutions
and then collectively discuss them.
Does this work? Does this
always work? Are there more
efficient solutions? And the like. By
the end of the class, all students were aware of the best approach(s) and why
they worked. Not, ÒAll solutions are equally valuedÓ, nor the instructor
learning along with his students as is often advocated in the US; the
instructor was in complete command.
Although the classes were warm and a little noisy, they were focused on
the topic and the instructor was clearly in charge, a ÒSage on the StageÓ, even
if the stage might have been anywhere in the room. Why that was not Mr. TakahashiÕs
experience I have no idea, and maybe it was not, but my guess is he has learned
the US education industryÕs rhetoric rather than JapanÕs historic reality. NYUÕs Alan Siegel provided a nice
summary of these videotapes:

http://www.cs.nyu.edu/faculty/siegel/ST11.pdf

While
we are on the subject of the traditional Japanese approach, StiglerÕs tapes
provided concrete evidence of how far off US math ed ÒreformersÓ are missing the right ideaÉ One of the
very active projects on teaching mathematics the NCTM way was UCLAÕs
well-funded Mathematics Renaissance.
With all of the publicity surrounding the 1995 *TIMSS*, some states - even a few large school districts - contracted
with StiglerÕs group to do video studies of their own programs for comparison
with the Japanese, German, and ÒordinaryÓ US schools. There was a conviction of math reformers
that they were doing things Òthe Japanese wayÓ (of course, not TakahashiÕs recollection
thereof!) and Mathematics Renaissance had enough federal money to have them
study and compare some Mathematics Renaissance classrooms with the
Japanese. The report was
enlightening for those with their eyes open (not the reformers, unfortunately). The December [1997] issue of the *Communicator* of the California
Mathematics Council (think California chapter of the NCTM) had an article by
two of its leaders, Nanette Seago and Judy Mumme, entitled "Mathematics Renaissance *TIMSS* Study" that describes the
project. Instead of coming closer
to emulating the high performing Japanese, the Mathematics Renaissance program
had been encouraging a pedagogical philosophy that was even further from the
Japanese model than were typical U.S. classrooms.

"The amount of time that teachers engage in
teacher/demonstration was analyzed.
Japanese lessons not only contain more Teacher Talk/Demonstration than
the other three groups, but more time is devoted to it when it does occur. In the case of Mathematics Renaissance
lessons, such segments never occurred.
There are dramatic differences between the percent of lessons in which
Teacher Talk/Demonstration occur, 64% of the Japanese lessons and 0%
Renaissance lessons."

Other
evidence that StiglerÕs tapes showed about JapanÕs classrooms (along with his
1993 book with Harold Stevenson, ÒThe Learning GapÓ) was
that calculators were never seen in Asian elementary (K-5?) school classrooms
and routine calculations were performed routinely and accurately. Perhaps Mr. Takahashi has been able to
fix that ÒproblemÓ in Japanese schools to emulate our own ÒsuccessÓ?

That
was an exemplary model of NCTM-style classrooms but the math reform philosophy
had been strong well before that.
In fact, the 1989 NCTM Standards represented more of an official
confirmation of efforts that had been underway for more than a decade so even
the 1995 *TIMSS* was not comparing
traditional Òdirect instructionÓ American mathematics classrooms (curricula and
pedagogy) with schools of other nations, many of those schools were already
well on their way and the infamous Math Wars had begun. Many of the controversial curricula were
not striving for automaticity of the number facts and instead of teaching the
traditional algorithms of arithmetic Òmy groupÓ was (and is!) expected to
explain our method to Òyour groupÓ with the justification mathematically
tenuous at best. Better to be told
(even though many forget they were ever told!) than the blind leading the
blind. Those videotapes were
helpful in that regard as well.
Stigler and Hiebert described the situation of
genuine mathematical support of the process under study strikingly, or may be
shockingly, similar to the percents at the
Mathematics Renaissance schools, 0% of the US schools spent time on mathematically
supportable proof.

James
W. Stigler and James Hiebert, Phi Delta Kappan 79,
No. 1 (September 1997) 14-21:

ÒOne feature on which the team members focused was
deductive reasoning, a form of mathematical activity that they considered
central for students' engagement in important mathematics. They defined
deductive reasoning as the reasoning needed to draw logical conclusions from
premises. Mathematical proofs are the most familiar form of such reasoning.
Deductive reasoning, as defined by the Math Group, was not common. Only
one-fourth of the 90 lessons contained instances of it. As it turned out, these
instances were found in 62% of the Japanese lessons, 21% of the German lessons,
and 0% of the U.S. lessons.Ó

Relevant
to the Mathematics Renaissance situation described earlier, one of the precepts
of US-style Òreform mathÓ is avoidance of direct teaching (ÒlecturingÓ is a
dirty concept if not a dirty word).
Although still far short of the Japanese instructional quality, ordinary
US classrooms were observed doing better than the Mathematics Renaissance
classrooms:

ÒAlthough it is true that Japanese teachers give students
time to struggle with challenging problems, they often follow this up with
direct explanations and summaries of what the students have learned. This is
why Japanese teachers were coded as engaging in more direct lecturing than
either German or U.S. teachers. Although the time devoted to lecturing was
minimal in all three countries, 71% of Japanese lessons contained at least some
lecturing, compared with only about 15% of German and U.S. lessons. Japanese
teachers also control the direction of the lesson in subtle ways, such as
creating conditions in the classroom that will govern the kinds of solution
methods students are likely to invent. For example, to begin a lesson, they
often select problems that can be solved by modifying methods that were
developed during the previous lesson.Ó

DonÕt
miss the clear necessity of well-qualified classroom leadership.

************************************************

Back
to TakahashiÕs mistaken description of what was developing in Japan (exemplary
teaching was already there!):

ÒOver the next 12 years, as the Japanese educational system
embraced this more vibrant approach to math,Ó

There
is some historical fact in his description of moving toward the NCTM model of
the blind leading the blind (instead of top-quality mathematics classrooms as
documented). This was due to the
growing professional mathematics education (as opposed to the discipline of
mathematics) influence in Japan but well before the end of the 2000s falling
performance for university preparedness and international comparisons had
created quite a stir - their own Math Wars:

http://www.nippon.com/en/in-depth/a00601/

ÒThis time protest would come from universities. And rather
than educational researchers or arts instructors, it was the math and science
instructors who were speaking out.Ó

Specific
implications for current Japanese classrooms can be seen in an interesting, if little-known, late 2000s study of careful observation of
representative middle schools in Japan conducted by Linfield College. The study indicate that these classrooms
are highly reminiscent of my own precollegiate
experience and the way we usually conduct university-level courses;
announcement and summary : http://www.linfield.edu/linfield-news/students-and-prof-examine-why-japan-outsmarts-u-s-in-math-scores/

ÒStudents sit in rows and are expected to listen quietly.
Teachers rely on direct instruction rather than investigative mathematics, but
although they ask few questions, the questions they do ask are useful in
guiding student understanding.Ó

Given
the heavy reliance on calculators in the early grades and graphing calculators
later on in American-style Òreform mathÓ (coinciding with downplaying of competence
with the standard algorithms of elementary school mathematics), the following
is enlightening:

The biggest surprise was a shocking lack of technology in
Japanese classrooms. ÒNot a single student pulled out a calculator during
class,Ó Drickey said. There were no overhead
projectors, televisions, computers or laptops.

ÒBut lack of reliance on technology may lead to higher
scores for Japanese students,Ó she said. ÒThe ability to think mathematically,
without the aid of an outside source, could help students process mathematical
problems more accurately and efficiently.Ó

Unfortunately,
I do not have an easy referral reference to the full school-by school data but
I will provide it on request.

************************************************

ÒIt 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.Ó

There
are several levels of misconception involved here. Perhaps the most important is that the
New Math of the late 50s and 60s was incomparable with Òreform mathÓ embodied
in the *NCTM Standards*. I was in college and a young high school
mathematics teacher (inner suburb of Chicago) so well aware of that reform
movement and its practicality or lack thereof in a real-world school. I was well aware of, and liked, the
iconic curricula of the School Mathematics Study Group (universally abbreviated
SMSG) but it never could have worked at my high school. A large part of the problem was that
reported Òinferiority complexÓ had convinced some mathematics education leaders
(who were then highly mathematically competent) that US mathematics needed more
formality, almost the antithesis of Òmy group has decidedÓ of the modern
ÒconstructivistÓ era. In fact, some
of the New Math curricula was TOO formal for almost
every high school student, not just the teachers who were generally
mathematically competent (for the mathematically initiated, think Bourbaki). An
example of that would be Max BebermanÕs curriculum of
the University of Illinois Committee on School Mathematics (UICSM). It purported to be Òdiscovery learningÓ
but it was only in the sense of (again, for the mathematically initiated) the
ÒMoore MethodÓ of R. L. Moore.

SMSG
was workable at some places and did have positive influences on curricular
development but from its inception it could not have work generally. Interestingly, the seminal meeting for
SMSG was held at Stanford University and the ÒgodÓ of discovery learning
in problem solving situations, George Polya, (his *How to Solve It* books are, by some,
almost taken as divine writing) was in residence and strongly encouraged to
participate. He wanted nothing to
do with the project because he was sure it was a failed effort before it ever
started. Great
for a select few and a train-wreck in the making for the masses in ordinary
classrooms with ordinary students and ordinary instructors.

There
was a more fundamental problem with the Òneed forÓ the New Math, however;
inadequate mathematics instruction was not the reason for the Òinferiority
complexÓ. In fact, long before
newly educated math-based scientists were having an influence, the US had long
since left the USSR and the rest of the world in the dust. Those same Òpoorly math educatedÓ
scientists and technicians had proved that there was no reason for a feeling of
inferiority. IÕm glad that the NSF
(National Science Foundation) spent a lot of money educating me - and I would
like to see more of the same for mathematically talented students now - but
Sputnik only provided the political impetus, not a genuine necessity. From a one-room country school through a
small Iowa high school, I got the mathematics knowledge base that I needed and
so did several others in my small class.
Moreover, that situation was highly representative of the nation. We were getting a much higher percentage
of students genuinely prepared for (I hate the term) STEM careers. Too much drill-and-kill arithmetic in the early years? Certainly, and I hated it, but there
were plenty of word problems that were motivating and the right seeds had been
planted. In those terrible bubble-in-your-answer
state tests (the Iowa Tests of Basic Skills), I consistently busted the top in
spite of those awful teachers and awful math curricula. Sarcasm switch off - a modest reform
would have been reasonable but the attempt for a revolution was unwarranted.

************************************************

ÒThe 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.Ó

Why
are we not surprised? These
infamous street urchins of Recife in mathematics education literature being
recycled yet again?! How long has it been now, a couple decades at least, maybe three? If it werenÕt for schools (and too many
donÕt have access to good ones) these kids would be both illiterate and
innumerate. Able
to sell seashells by the seashore and get the money right? Certainly. Math-based career opportunity because of
this street experience? Well...

ÒWhen
cognitive scientists presented the children with the very same problem,ÉÓ

What
real cognitive scientists would do is try to somehow measure future potential
because of this unusual street experience.
My guess is that a well-run (lots are NOT) HeadStart
experience followed by a good elementary school math program would be much
better academic preparation than a decade or more trying to scratch out a
living on the streets of Recife.

************************************************

ÒLampert 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.Ó

What a
ridiculous waste of zero-sum math time, especially in 5^{th} grade with
a 3^{rd}, or at most 4^{th}, grade concept. Take 5 or 10 minutes, if necessary,
clearly review the concept and move on!

ÒA 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."

Confidently
declared or no, as a stand-alone statement, it is pure nonsense - the number 12
doesnÕt have ÒsectionsÓ although it is easy to read into it what he meant. Calling it a ÒproofÓ is ridiculous
but it is informative. It probably
means that Ms. Lampert never understood Òsophomore
geometryÓ or any other mathematical proof.
It amazes me that the last generation or two of professional mathematics
education has been able to do what a couple of millennia was not able to do;
kill the greatest gift of the ancient Greeks to modern thought, semi-formal
deductive logic. Sin has many
forms. As we saw from the Stigler
and Hiebert observations, the Japanese schools knew
better and know better.

************************************************

ÒWith 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.Ó

This
is all true and it is an insurmountable problem. These people are Òin the pipelineÓ,
often with tenure but at the very least with an implied Òproperty rightÓ having
served for years with glowing evaluations.
Their mathematics competence will never be at the level needed to
conduct such an environment effectively; it is very hard to Òcarry it offÓ well
with good mathematics skills and teaching talent let alone a lifetime of
(understandable) math avoidance.
The part that is not understandable is how we have long since ignored
mathematics competence among teachers and administrators. Beyond that insurmountable problem is
the surmountable one that professional math ed refuses to face, independent and objective
assessments of (borrowing a term from medicine) efficacy and toxicity; i.e.,
potentially negative side effects.
Math ed ÒresearchÓ
and education ÒresearchÓ, in general, does not come close to reputable
standards of research. The medical
admonition, ÒFirst Do No HarmÓ, needs to be addressed and it has not been for
the *CCSS-M*.

ÒTextbooks,
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.Ó

This
is entirely accurate and it will remain so. As opposed to genuinely good standards
(the California Mathematics Content Standards are great example), there is so
much pedagogical fuzz deliberately embedded in the *CCSS-M* that such an evaluation is inherently, and inconsistently,
subjective. Unfortunately, Mr. Daro
himself is representative of the inherent problem of math education in the US. In spite of his long history of leading
NCTM-style mathematics education reform in California and across the nation,
his degree is a bachelors degree in English, not mathematics, not even a
math-based academic discipline.
That situation is pervasive throughout the professional education
behemoth. Moreover, he comes with
heavy baggage as the primary author of the mathematics portion of an earlier
effort to reform education standards, The ÒNew Standards ProjectÓ, and as the
principal author of its associated assessments, the *New Standards Mathematics Reference Exams* (*NSMRE*). Using the full
data of the Pittsburgh Public Schools, StanfordÕs Jim Milgram and myself
confirmed that the alleged enlightened results of the *NSMRE* were completely meaningless; worse,
misinforming. Using exactly the
same cohort of students, 4^{th} grade in 2001 with the *NSMRE* and 5^{th} grade in 2002
with the *PSSA*, the regular state exam
for Pennsylvania, the highly touted - and NSF funded - math program of Pittsburgh was shown
to be no more effective, especially with minority children from with low
socioeconomic and low education level backgrounds, than other attempts have
been there and elsewhere. Indeed,
the data followed by on-site visitation, confirmed our suspicions; the
objectively-assessed more successful schools had developed ways to compensate
for the program or, in the best case, the school had purchased an entirely
different math curriculum (much more traditional in nature) to be used whenever
the MPs (Math Police) were out of sight.

What
about the *NSMRE* versus systemwide
exams? This traditionally
low-performing cohort of students who, in 4^{th} grade, had performed
collectively a little above the national norm for ALL students were shown to
really have been no better educated mathematically than the ugly
pre-suppositions would have suggested.
One year later, using the same *CCSS-M*
friendly curricula and pedagogy, some 60% of these students were performing in
mathematics Below Basic or Far Below Basic; lifeÕs handwriting already on the
wall. (Data available on request.)

Not
only was Mr. DaroÕs influence very strong in writing
the *CCSS-M* (along with lead writers
Jason Zimba and William McCallum), his New Standards Project experience does
not bode well for the new-shiny assessments to go with the *CCSS-M*, especially those states with exams being prepared by SBAC,
the Smarter Balanced Assessment Consortium. He is also in charge of the mathematics
assessment therein. Not only does
his instrumental involvement with the *NSMRE*
offer reasonable concern, he was also actively involved in the other forerunner
of these ÒnewÓ assessments, the *MARS
Balanced Assessments*. In spite
of lots of NSF and Noyce Foundation money, these
exams are just as Òoff-the-wallÓ as are the *NSMRE*.

************************************************

ÒGuiding
the student through the exercise <snip> 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."

Addressing
another observation of Magdalene Lampert, her
reasonable assessment of this limitation of so-called discovery or
constructivist pedagogy should not go unnoticed. In fact, I have observed several
elementary school math classes doing exactly that; elaborately developed
student small-group activities where students were to ÒdiscoverÓ some important
mathematical underpinning.
Problem? I was the only
person in the room, teacher included, who understood the activity. The activity took place including
expensive ÒmanipulativesÓ, was completed and recorded
as instructed, and neatly put away.
Math lesson over with nothing accomplished except for babysitting.

************************************************

ÒBy 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, YÕall, 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.Ó

Accurate
as this may be, we seem to have a chronological problem. 1995 is almost two decades ago and the
NCTM Standards had only been out for 6 years (1^{st} edition in 1989)
so the reported pervasive influence in Japan could not yet have had that
influence. As the videotapes of the
Japanese classrooms confirmed, their highly competent mathematics practices were
already well ingrained and videotapes of a decade earlier would not have been
all that different. One of the
observations about the Japanese classrooms in comparison with the American ones
in this Òspoke more oftenÓ comment is that the Japanese classrooms were
generally focused on the problem at hand, noisy for the sake of mathematics not
for the sake of noisy. Seldom is
that the norm in US classrooms but, if conditions are right, it is a delightful
learning environment. In fact, I
experienced it in one of my own high school classes, a senior-level class that
the math teacher had been able to persuade the powers-that-be to offer for a
few of us (5 or possibly 6) with strong math-based backgrounds as an
introduction to university-level mathematics where we were all headed the
following year. It was great. Joshing each other, including the
instructor, helped keep the atmosphere relaxed but it was never malicious and
always focused on the mathematics we were studying. DonÕt try that with 30-some students,
many ill-prepared for the class and most of them required to be there against
their will, and ÒledÓ by a teacher also ill-prepared for the class.

************************************************

ÒFinland, 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.Ó

A
little-known truth is that FinlandÕs highly publicized jump to near the
top-in-the-world math performance was only by education industry
smoke-and-mirrors. FinlandÕs
performance on the *TIMSS* (now
ÒTrendsÓ because of its strong reputation but ÒThirdÓ was in 1995) did not jump
dramatically as casual observers would conclude from all of the publicity. No, it was based on using an entirely
different international assessment.
*PISA* (*Programme for International Student Assessment*) is an exam prepared
by the Freudenthal Institute with lots of US math ed ÒinsightÓ directing the process
and it is an invalid indicator of mathematics competence. The fact is that FinlandÕs actual
mathematics competence has not improved.
Updated *TIMSS*
results exist and confirms that FinlandÕs performance, as of 2011,
remains so-so, very comparable to that of the US and far behind those Asian
nations:

http://timss.bc.edu/TIMSS2011/index.html

http://timss.bc.edu/TIMSS2011/downloads/T11_UserGuide.pdf

Chapter
2, P 23 for 4th grade (Singapore still on top, Finland slightly behind US)

Chapter 3, P 57 for 8th-grade (Korea on top
followed closely by Singapore, Finland slightly ahead of US)

__Note__: Although still high, officials in Japan were chafing at
not being 1^{st} or 2^{nd}.

************************************************

ÒOf 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.Ó

Rest
assured, it is not the most important lesson - it is not even close. By far the most important is what Japan,
Korea, Singapore, and lots of other high-performing nations have - something
that used to be a given in the US but we have long since lost - mathematics
competence of the teaching faculty, especially at the elementary level. In spite of its so-so performance, even
Finland is considerably better off than we are in that regard; most countries
are. Not only does the published
research of the professional education industry fail to meet general academic
standards, candidates for teaching are, generally speaking, poorly prepared
coming in to college and weakly educated while there with grade inflation in
professional education courses at a ridiculous level. Objective assessments of prospective
teachers (primarily through the SAT and ACT) and those pursuing higher degrees,
often in education administration, (primarily through the GRE) are consistently
in the lowest quartile of college students. By contrast, prospective teachers in
Japan are consistently in the top quartile and many from the top 10%. Jim Stigler (of the *TIMSS* video project and *The Learning
Gap*) reported about an education PhD candidate from Japan that he met at a
well-known university who was here in the US for a very enlightening
reason: On completion of her preservice education to become an elementary school teacher
in Japan, she had failed the rigorous competency exam required to become a
teacher (think bar exam in the US).
Rather than wait the mandatory year and do the prerequisite ÒcrammingÓ
to take the test again, she accepted an offer to pursue a PhD in education here
in the US.

ÒThe 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.Ó

Exactly my point. Moreover, expecting the professional
education industry to change the status quo is like asking the leopard to
change its spots. The heart of the
problem is the so-called ÒprofessionalÓ education emanating from our schools
and colleges of education. It is
worse than the blind leading the blind - at least the blind would try - the
education industry has a strong vested interest in the status quo. No change is going to come from within
and its tentacles reach into legislatures that control credentialing
requirements that would have to be dramatically modified in order to effect
real change. It will not happen.

Beyond
the problems that impede substantial progress that were mentioned, there is
another that is true in any academic area but overwhelmingly important in
mathematics, adequate preparation of the students. The exciting kind of classroom of the
Japanese videotapes (or of my own senior-level class that I mentioned) requires
that the students all be engaged, at least mentally if not vocally. That is simply impossible in the US
ÒidealÓ - often legally mandated - heterogeneous classrooms. Mathematically weak students Òturn offÓ
very early, sometimes before the class even begins. At the other end of the spectrum, 5^{th}
graders ÒprovingÓ that 4×6 = 24 by making 4 rows with 6 stars in each
row, or some such, is ridiculous.
It is an appropriate discussion item in 2^{nd} or 3^{rd}
grade but it is educational idiocy that should badly irritate a well-prepared
fifth-grader. Slightly more
sophisticated, but still inappropriate at that level would be ÒprovingÓ that
4×6 = 6×4 by turning the paper 90¡. Have 3rd-grade automaticity well in hand
- among everybody - and move on. In US classrooms, too often that is
totally unrealistic. Fortunately,
at the university level (and US universities remain first-rate), we are not
burdened by such ed industry
insight where thousands of ÒresearchÓ publications allegedly ÒproveÓ that
heterogeneous classrooms are best for everyone instead of the diametric
opposite.

An excellent summary/admonition of what the nation should
expect from all of the hoopla and funding, both private and public, trying to
prop up the *CCSS-M* was given in that
1997 *Kappan* paper by Stigler and Hiebert:

Ò**Beware of Simple
Solutions**

Given the high mathematics achievement of Japanese
students, it is tempting to conclude that U.S. teachers should teach more like
their Japanese counterparts. Although there are probably many useful ideas for
U.S. classrooms in the Japanese videos, we are pessimistic that such ideas can
simply be imported. Indeed, if teaching could be changed by just disseminating
ideas, the record of reform in the U.S. would be more successful than it is.Ó

Wayne
Bishop, PhD

Professor
of Mathematics

California
State University, LA

Email:
wbishop@calstatela.edu

************************************************

Some
Relevant Background:

1988-90 Math Advisory
Panel to CAÕs Credentialing Commission

1995 SuperintendentÕs
Math Task Force (CA first addressing the Math Wars)

1996 Invited
Testimony to CA Assembly (regarding tentative education legislation)

1999 CRP
(Content Report Panel for assessment of math curricula against *CA Mathematics Content Standards*)

2001 CRP
(see preceding - after publishers had had legally mandated time for writing)

2002 Commissioned
by Pittsburgh Public Schools to evaluate its math program (along with R. James
Milgram, Philip Daro, and Uri Treisman - two critical
and two supportive)

2005 CRP Interim Approval (an
opportunity for publishers to resubmit)

__Note__: Some math curricula that did not make the California CRP
approval in 1999 and/or 2001 was approved by the state
in 2007 under the same *CA Mathematics
Content Standards*. How come the
inconsistency? Many statesÕ standards, the *NCTM
Standards*, and the *CCSS-M* include
enough confounding pedagogy and lack of specificity that objective assessment
of consistency is inherently impossible.
The result, of course, is that all publishers point out how consistent
their product is with whatever standards are in play. By contrast, the *CA Mathematics Content Standards* are quite clear in what should be
considered to be grade-level compatible with them. How can a submission have been turned
down in 2001 but approved in 2007 under the same standards with only cosmetic
changes in content and pedagogy?

The answer has to do with the highly unusual nature of the
1999 and 2001 CA approval process that was already changing by 2005 and fully
in place by 2007. The first CRP
were very unusual in that degrees in mathematics education did not qualify for
membership, mathematics only, and were given Òtrump rightsÓ over the rest of
the approval process. Even if we
liked a particular curricular submission, we were obligated to reject it if it
didnÕt pass muster against the *CA
Standards*. Moreover, our
decision was final. If it didnÕt
meet the CA Standards, STOP. By
2007, there was no independent CRP assessment, only one CRP member on each
evaluating sub-team of the curriculum committee at large. Democracy ruled, one
person one vote, and some math curricula that did not meet CRP approval
- and received a ÒnoÓ vote - was approved for use in California. *CÕest** la vie.*

Access this
essay in .pdf format

Recommended Citation:

Bishop,
W. (2014). Why Do Americans Stink at Math?
Some of the Answer. *Nonpartisan
Education Review / Reviews, 10*(1). http://nonpartisaneducation.org/Review/Reviews/v10n1.pdf