The Washington Post

May 31, 2005


10 Myths (Maybe) About Learning Math


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" (  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 or to the NCTM at  The NCTM Web site is, 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

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



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

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



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.