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://email@example.com or to the NCTM at
http://firstname.lastname@example.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
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
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
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
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
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
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
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
Myth #9 -- NCTM math reform reflects the programs and practices in higher
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
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
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.