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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