The Washington Post

May 31, 2005

By Jay Mathews

Washington Post Staff
Writer

I love debates, as
frequent readers of this column know. I learn the most

when I am listening to
two well-informed advocates of opposite positions

going at each other.

I have held several
debates here, although not all of them have worked

because the debaters
lose focus. One will make a telling point, and the

other, instead of
responding, will slide off into a digression.

So when I found a new
attack on the National Council of Teachers of

Mathematics (NCTM),
the nation's leading association for math teachers, by a

group of smart
advocates, I saw a chance to bring some clarity to what we

call the Math Wars.
For several years, loosely allied groups of activist

teachers and parents
with math backgrounds have argued that we are teaching

math all wrong. We
should make sure that children know their math facts --

can multiply quickly
in their heads and do long division without

calculators, among
other things -- or algebra is going to kill them, they

say. They blame the
NCTM, based in Reston, Va., for encouraging loose

teaching that leaves
students to try to discover principles themselves and

relies too much on calculators.

The NCTM people, on
the other hand, said this was a gross misstatement of

what they were doing.

The advocates call
their new assault "Ten Myths About Math Education and Why

You Shouldn't Believe
Them" (http://www.nychold.com/myths-050504.html.) I took the

myths, and their
explanation of each, and asked the NCTM to respond to each

one. Here is the
result. There are some quotes that are not attributed, but

are found in sources
cited on the myth Web page, and some technical

language, but I think
this provides a good quick review of what this raging

argument is all about.

Feel free to send your
comments to one of the people who came up with the

list of 10, Elizabeth
Carson at http://nycmathforum@yahoo.com
or to the NCTM at

http://president@nctm.org. The NCTM Web site is

http://www.nctm.org/about/position_statements,
and the names of the

dissident group are on
the myth Web page.

Myth #1 -- Only what
students discover for themselves is truly learned.

Advocates: Students
learn in a variety of ways. Basing most learning on

student discovery is
time-consuming, does not insure that students end up

learning the right
concepts, and can delay or prevent progression to the

next level. Successful
programs use discovery for only a few very carefully

selected topics, never
all topics.

NCTM: NCTM has never
advocated discovery learning as an exclusive or even

primary method of
instruction. In fact, we agree that students do learn in a

variety of ways, and
effective learning depends on a variety of strategies

at appropriate times.
The goal is not just to know math facts and procedures

but also to be able to
think, reason and apply mathematics. Students must

build their skills on
a strong foundation of understanding.

Myth #2 -- Children
develop a deeper understanding of mathematics and a

greater sense of
ownership when they are expected to invent and use their

own methods for
performing the basic arithmetical operations, rather than

being taught the
standard arithmetic algorithms and their rationale, and

given practice in
using them.

Advocates: Children
who do not master the standard algorithms begin to have

problems as early as
algebra I.

The snubbing or
outright omission of the long division algorithm by NCTM-

based curricula can be
singularly responsible for the mathematical demise of

its students. Long
division is a pre-skill that all students must master to

automaticity for
algebra (polynomial long division), pre-calculus (finding

roots and asymptotes),
and calculus (e.g., integration of rational functions

and Laplace
transforms.) Its demand for estimation and computation skills

during the procedure
develops number sense and facility with the decimal

system of notation as
no other single arithmetic operation affords.

NCTM: NCTM has never
advocated abandoning the use of standard algorithms.

The notion that NCTM
omits long division is nonsense. NCTM believes strongly

that all students must
become proficient with computation (adding,

subtracting,
multiplying, and dividing), using efficient and accurate

methods.

Regardless of the
particular method used, students must be able to explain

their method,
understand that other methods may exist, and see the

usefulness of
algorithms that are efficient and accurate. This is a

foundational skill for
algebra and higher math.

MYTH #3 -- There are
two separate and distinct ways to teach mathematics.

The NCTM backed
approach deepens conceptual understanding through a problem

solving approach. The
other teaches only arithmetic skills through drill and

kill. Children don't
need to spend long hours practicing and reviewing basic

arithmetical
operations. It's the concept that's important.

Advocates: "The
starting point for the development of children's creativity

and skills should be
established concepts and algorithms. ..... Success in

mathematics needs to
be grounded in well-learned algorithms as well as

understanding of the
concepts."

What is taught in math
is the most critical component of teaching math. How

math is taught is
important as well, but is dictated by the "what." Much of

understanding comes
from mastery of basic skills -- an approach backed by

most professors of
mathematics. It succeeds through systematically

empowering children
with the pre-skills they need to succeed in all areas of

mathematics. The myth
of conceptual understanding versus skills is

essentially a false
choice -- a bogus dichotomy. The NCTM standards

suggested "less
emphasis" on topics needed for higher math, such as many

basic skills of
arithmetic and algebra.

"That students
will only remember what they have extensively practiced --

and that they will
only remember for the long term that which they have

practiced in a
sustained way over many years -- are realities that can't be

bypassed."

NCTM: Math teaching
does not fall into two extremes. There are several ways

to teach effectively.
Even a single teacher isn't likely to use the same

method every day. Good
teachers blend the best methods to help students

develop a solid
understanding of mathematics and proficiency with

mathematical
procedures.

It's worth noting that
standard algorithms are not standard throughout the

world. What is most
important is that an algorithm works and that the

student understands
the math underlying why it works.

Every day teachers make
decisions that shape the nature of the instructional

tasks selected for
students to learn, the questions asked, how long teachers

wait for a response,
how and how much encouragement is provided, the quality

and level of practice
needed -- in short, all the elements that together

become the
opportunities students have to learn. There is no

one-size-fits-all
model.

Myth #4 -- The math
programs based on NCTM standards are better for children

with learning
disabilities than other approaches.

Advocates: "Educators
must resist the temptation to adopt the latest math

movement, reform, or
fad when data-based support is lacking. ....."

Large-scale data from
California and foreign countries show that children

with learning
disabilities do much better in more structured learning

environments.

NCTM: Most of the math
programs published in this country claim to be based

on the NCTM Standards.
More important than the materials we use is how we

teach. Students, all
students, are entitled to instruction that involves

important mathematics
and challenges them to think.

Myth #5 -- Urban
teachers like using math programs based on NCTM standards.

Advocates: Mere
mention of [TERC, a program emphasizing hands-on teaching of

math that this group doesn't
believe demands enough paper and pencil work]

was enough to bring a
collective groan from more than 100 Boston Teacher

Union representatives.
..... "

NCTM: Curricular
improvement is hard, takes a lot of work, and demands

support -- for the
teacher, for students, and for parents. It should be

noted that Boston
students using the TERC-developed curriculum seem to be

thriving. The
percentage of failing students on the Massachusetts state

assessment decreased
from 46 to 30 percent and students scoring at the

Proficient and
Advanced categories increased from 14 to 22 percent between

2000-2004 (Boston
Globe, December 14, 2004).

Myth #6 --
"Calculator use has been shown to enhance cognitive gains in

areas that include
number sense, conceptual development, and visualization.

Such gains can empower
and motivate all teachers and students to engage in

richer problem-solving
activities." (NCTM Position Statement)

Advocates: Children in
almost all of the highest scoring countries in the

Third International Mathematics
and Science Survey (TIMMS) do not use

calculators as part of
mathematics instruction before grade 6.

A study of calculator
usage among calculus students at Johns Hopkins

University found a
strong correlation between calculator usage in earlier

grades and poorer
performance in calculus.

NCTM: The TIMSS 1999
study of videotaped lessons of eighth-grade mathematics

teachers revealed that
U.S. classrooms used calculators significantly less

often than the
Netherlands (a higher achieving country) and not

significantly
differently from four of the five other higher-achieving

countries in the
analysis. When calculators are used well in the classroom,

they can enhance
students' understanding without limiting skill development.

Technology (calculator
or computer) should never be a replacement for basic

understanding and
development of proficiency, including skills like the

basic multiplication
facts.

Myth #7-- The reason
other countries do better on international math tests

like TIMSS and PISA is
that those countries select test takers only from a

group of the top
performers.

Advocates: On NPR's
"Talk of the Nation" program on education in the United

States (Feb. 15,
2005), Grover Whitehurst, director of the Institute of

Education Sciences at
the Department of Education, stated that test takers

are selected randomly
in all countries and not selected from the top

performers.

NCTM: This is a myth.
We know that students from other countries are doing

better than many U.S. students,
but certainly not all U.S. students. One

reason U.S. students
have not done well is that the way we have taught math

just doesn't work well
for enough of our students, and we have the

responsibility to
teach them all.

Myth #8 -- Math
concepts are best understood and mastered when presented "in

context"; in that
way, the underlying math concept will follow

automatically.

Advocates:
Applications are important and story problems make good

motivators, but
understanding should come from building the math for

universal application.
When story problems take center stage, the math it

leads to is often not
practiced or applied widely enough for students to

learn how to apply the
concept to other problems.

"[S]olutions of
problems ..... need to be rounded off with a mathematical

discussion of the
underlying mathematics. If new tools are fashioned to

solve a problem, then
these tools have to be put in the proper mathematical

perspective. .....
Otherwise the curriculum lacks mathematical cohesion.

NCTM: For generations,
mathematics was taught as an isolated topic with its

own categories of word
problems. It didn't work. Adults groan when they hear

"If a train
leaves Boston at 2 o'clock traveling at 80 mph, and at the same

time a train leaves
New York ..... " Whatever problems and contexts are

used, they need to
engage students and be relevant to today's demanding and

rapidly changing
world.

An effective program
lets students see where math is used and helps students

learn by providing
them a chance to struggle with challenging problems. The

teacher's most
important job in this setting is to guide student work

through carefully
designed questions and to help students make explicit

connections between
the problems they solve and the mathematics they are

learning.

Myth #9 -- NCTM math
reform reflects the programs and practices in higher

performing nations.

Advocates: A recent
study commissioned by the U.S. Department of Education,

comparing Singapore's
math program and texts with U.S. math texts, found

that Singapore's
approach is distinctly different from NCTM math "reforms."

Also, a paper that
reviews videotaped math classes in Japan shows that there

is teacher-guided
instruction (including a wide variety of hints and helps

from teachers while
students are working on or presenting solutions).

NCTM: The study
commissioned by the U.S. Department of Education comparing

Singapore's
mathematics program and texts with U.S. math texts also found

that the U.S. program
"gives greater emphasis than Singapore's to developing

important 21st-century
mathematical skills such as representation,

reasoning, making
connections, and communication. The U.S. frameworks and

textbooks also place
greater emphasis on applied mathematics, including

statistics and probability."

NCTM's standards call
for doing more challenging mathematics problems, as do

programs in Singapore,
Japan and elsewhere, but they also recognize the

needs of 21st-century
learners.

Myth #10 -- Research
shows NCTM programs are effective.

Advocates: There is no
conclusive evidence of the efficacy of any math

instructional program.

Increases in test
scores may reflect increased tutoring, enrollment in

learning centers, or
teachers who supplement with texts and other materials

of their own choosing.
Also, much of the "research" touted by some of the

NSF programs has been
conducted by the same companies selling the programs.

State exams are
increasingly being revised to address state math standards

that reflect NCTM
guidelines rather than the content recommended by

mathematicians.

NCTM: True, there is
no compelling evidence that any curriculum is effective

in every setting, nor
are there data to show exactly what causes improvement

in student learning
when many factors are involved. There is evidence that

some of the more
recently developed curricula are effective in some

settings. However, the
effectiveness with which a program, any program, is

implemented is
critical to its success, as are teacher quality, ongoing

professional
development, continuing administrative support, and the

commitment of
resources. Again, the issue of effectiveness is more likely to

be attributable to
instruction than to any specific curriculum.

Contrary to what is
stated in some of these myths, there is no such thing as

an "NCTM
program." NCTM does not endorse or make recommendations for any

programs, curricula,
textbooks, or instructional materials. NCTM supports

local communities
using Principles and Standards for School Mathematics as a

focal point in the
dialogue to create a curriculum that meets their needs.

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