Memorize Multiplication Facts?
2005 Update Notes:
This Essay Deals with a Major Fallacy of NCTM Math:
Memorization and Practice are Ruled Out. So Kids Don't Remember
Specific Math Facts and Skills. But "Doing Math" Requires the Application
of Remembered Math Facts and Skills That Must First be Stored in the
The First Fallacy of NCTM Math is the Lack of
Genuine Math Content.
- Originally written in 1997, this essay refers to the 1989 version of the
NCTM Standards. But it's still relevant, because the 2000 version of the
NCTM Standards is based on the same constructivist philosophy.
- NCTM math is synonymous with "new-new math," "fuzzy math," and
On August 11, 1997 the New York Times published two Op-Ed pieces, one by
Lynne Cheney, former chair of the National Endowment for the Humanities, and the
other by Thomas Romberg, former chair of the Commission on Standards of The
National Council of Teachers of Mathematics (NCTM).
To view the Cheney and Romberg essays, click on the author's name below.
Cheney Described the "Constructivist" Philosophy of the NCTM
- "Students don't need to know multiplication tables in order to divide,
they say. Using objects and calculators they can figure it out - and thus
begin to create their own mathematical knowledge."
- "According to the council, stressing addition, subtraction, and worst of
all, memorization made students into passive receivers of rules and procedures
rather than active participants in creating knowledge."
- "The standards recommended that students get together with peers in
cooperative learning groups to "construct" strategies for solving math
problems, rather than sit in class with teachers instructing them."
- "Calculators were a necessity from kindergarten on, the council said,
because students liberated from "computational algorithms" could pursue
higher-order activities, like inventing personal methods of long division."
- "Although the council wants all students to learn to estimate and use
mental calculation and to be equally comfortable using paper and pencil,
calculators and computers, nowhere does it argue that students do not need to
memorize multiplication facts. Nor does the council say that students should
use a calculator for all computations."
- "Similarly, the examples the teachers council has produced often portray
students working on real-world problems, providing oral or written
explanations of how a problem was solved, or collaborating with other
students. But the council never meant to imply that such approaches are
appropriate for every lesson."
- "Sometimes even appropriate material is ineffective because the teacher
using it doesn't have enough math background or the right training"
- "It's unfair to attack the entire program because of initial missteps and
isolated examples of misapplied guidelines."
What Do the NCTM Standards Actually Say?Thomas Romberg suggests
that the NCTM Standards have been misunderstood. They don't explicitly rule out
all memorization, they don't recommend the constant use of calculators, and they
don't say kids must always work in groups, attempting to solve real-world
problems. We are to be comforted that the NCTM never intended an extreme
interpretation of their "less of this" and "more of that" recommendations. They
never said that "none" was the optimal form of "less", or that "always" was the
optimal form of "more". Apparently it's all the fault of poorly prepared
teachers. Of course, the answer is increased funding for NCTM-controlled teacher
Here's what the NCTM Standards actually say:
- First, and most importantly, the NCTM Standards discourage the
memorization of multiplication facts and any other specific math content.
Whenever memorization or memorization methods (practice) are mentioned, the
tone is always negative. Here are a few examples:
- "The curriculum should focus on the development of understanding, not on
the rote memorization of formula."
- "The 9-12 standards call for a shift in emphasis from a curriculum
dominated by memorization of isolated facts and procedures and by
proficiency with paper-and-pencil skills to one that emphasizes conceptual
understandings, multiple representations and connections, mathematical
modeling, and mathematical problem solving".
- "Classroom observations should gather information about whether
mathematics is portrayed as an integrated body of logically related topics
as opposed to a collection of arbitrary rules that students must memorize."
- Next, Thomas Romberg is literally correct, the NCTM Standards don't say
that "students should use a calculator for all computations". Here's
what they do say:
- "Contrary to the fears of many, the availability of calculators and
computers has expanded students' capability of performing calculations.
There is no evidence to suggest that the availability of calculators makes
students dependent on them for simple calculations. Students should be able
to decide when they need to calculate and whether they require an exact or
approximate answer. They should be able to select and use the most
- The clear message is that all kids, regardless of age, should decide
for themselves about the appropriate use of calculators. (So far, there
are no known instances of children who have personally decided to reject
the calculator in favor of the more difficult option of learning
- Finally, what do the NCTM Standards have to say about group learning?
- "This approach instills in students an understanding of the value of
independent learning and judgment and discourages them from relying on an
outside authority to tell them whether they are right or wrong."
- "Students should frequently work together in small groups to solve
problems.... They should verify their own thinking rather than depend on the
teacher to tell them whether they are right or wrong."
A Major Fallacy Behind NCTM MathThe
NCTM's constructivist math educators want easy, stress-free math, so they reject
memorization and practice and thereby severely limit the student's ability to
remember specific math facts and skills.
Without specific remembered knowledge,
students must regularly revisit shallow content and rely on general
content-independent skills, such as "draw a picture" or "make a
Traditionally, K-12 math is the first man-made
knowledge domain where American children build a remembered knowledge base of
domain-specific content, with each child gradually coming to understand hundreds
of specific ideas that have been developed and organized by countless
contributors over thousands of years. With teachers who know math and sound
methods of knowledge transmission, the student is led, step-by-step, to remember
more and more specific math facts and skills, continually moving deeper and
deeper into the structured knowledge domain that comprises traditional K-12
math. This first disciplined knowledge-building experience is a key
enabler, developing the memorizing and organizing skills of the mind, and
thereby helping to prepare the individual to eventually build remembered
knowledge bases relative to other knowledge domains in the professions,
business, or personal life. The ongoing
strength of our information-age economy depends fundamentally on a ready supply
of millions of knowledge workers who can learn to understand and extend
thousands of specific knowledge domains, from aeronautical engineering and
carpentry to piano tuning and zoology. Although the specific facts,
skills, and organizing principles differ from domain to domain, genuine domain
experts must necessarily remember a vast amount of specific information that is
narrowly relevant to their targeted knowledge domains, frequently without the
possibility of transfer to other domains.
If Traditional Content is Out, What's NCTM Math
The major subtopics are calculator skills, math appreciation, and, general,
content-independent "process" skills. For example, in his New York Times
Op-Ed piece (August 11, 1997), Thomas Romberg placed the spotlight on general,
content-independent skills when he wrote about the foundational role of the
"four general standards - problem solving, communication, reasoning, and
connection". Of course, space didn't permit him to give you the new-new
definitions of these high-sounding terms. Fortunately, we have space here. The
truth is in the details.
First, forget your old-old ideas:
Quoting from the NCTM Standards, here's the new-new way:
- New-new "problem-solving" doesn't depend on first learning specific math
- New-new "communicating" doesn't emphasize the correct use of the precise
symbols and language of math.
- New-new "reasoning" doesn't refer to the step-by-step application of
remembered math content.
- New-new "connecting" isn't about relating new math knowledge to previously
- The NCTM problem-solving strategies are:
- "Using manipulative materials"
- "Trial and error"
- "Making a list"
- "Drawing a diagram"
- "Looking for a pattern"
- "Acting out a problem"
- "Guess and check"
- Grade K-4 NCTM communication is about:
- "the integration of language arts as children write and discuss their
experiences in mathematics."
- "children might draw pictures and then tell or write stories about the
- "Students can write a letter to tell a friend about something they have
learned in mathematics class."
- Grade 9-12 teachers of NCTM communication are to:
- "Direct instruction away from a focus on the recall of terminology and
routine manipulation of symbols and procedures."
- Recognize that "in mathematics, just as with a building, all students
can develop an understanding and appreciation of its underlying structure
independent of a knowledge of the corresponding technical vocabulary and
- Understand that "technical symbolism should evolve as a natural
extension and refinement of the students' own language".
- NCTM reasoning is about:
- Helping "children learn that mathematics is not simply memorizing rules
and procedures but that mathematics makes sense, is logical, and enjoyable".
- "Creating and extending patterns of manipulative materials and
recognizing relationships within patterns".
- Recognizing that "most fifth graders are still concrete thinkers who
depend on a physical or concrete context for perceiving regularities and
- Understanding that "even the most advanced students at the 5-8 level
might use concrete materials to support their reasoning".
- Encouraging students "to explain their reasoning in their own words".
- Recognizing that, prior to grade 9, "experience both in and out of
school has taught them to accept informal and empirical arguments as
sufficient", (Note: "Informal arguments" refers to the use of concrete
manipulatives, and "empirical arguments" refers to generalizing from
- Under the heading of connections, the NCTM wants to:
- Insure that children "will not need to learn or memorize as many
procedures; and will have the foundation to apply, recreate, and invent new
ones when needed".
- Help students to "view mathematics as an integrated whole rather than as
an isolated set of topics".
- Insure that "as in the earlier grades, teachers in grades 9-12 should
introduce a new topic by exploring appropriate concrete
Brought To You by "Math Educators", Not MathematiciansWhen Thomas
Romberg tells you that the NCTM Standards have "received wide support from
mathematicians and math educators", he's hoping you're not too picky about the
definition of "mathematician". Sure, there are a few mathematicians who appear
to go along, but these typically haven't read the NCTM Standards and just assume
that they can't be that bad. (Of course, "a few" suffices as a "proof" for
Thomas Romberg, using the new-new math definition of "empirical reasoning".)
Math Educators Do Not Speak for Mathematicians
- Mathematicians reject the idea that you can "do math" without "knowing
- Genuine mathematicians know that a necessary condition for reasoning
mathematically is a remembered knowledge base of specific math facts and
specific math skills
- Mathematicians reject the early use of calculators, beginning in
kindergarten, and they question the current excessive use of calculators,
during all of the K-12 years.
- Mathematicians want parents to know that kids must be challenged, not
babied until grade 9 and beyond with concrete "manipulatives", and not
continually waiting until the child feels "ready".
- Mathematicians reject the substitution of "informal and empirical
arguments" for genuine mathematical reasoning and genuine mathematical proof.
- Mathematicians cringe at the belief that math appreciation (a cult-like
excitement about the "power" and "beauty" of math) is more important than
actually learning specific math content.
- Mathematicians reject the belief that general, content-independent skills
are in any way central to the process of learning mathematics.
Anti-Content Thinking Threatens All Americans
- If our kids never learn the importance of remembered knowledge, and if
they are programmed to think that memorization and practice are not necessary,
then what happens if they somehow reach medical school and need to quickly
memorize thousands of facts from Gray's Anatomy? It's difficult enough
even with the traditional preparation of the mind.
- This is not just about kids who go to medical school. The current
reigning educational philosophy is dangerous for all our children. If they
are to be successful in life, they must effectively use the amazing
knowledge-storing power of their brains. Are we really going to continue to
let today's educationists program our kids to believe that remembering
specific knowledge is a bad idea, and that computer "tools" and "look-up
skills" are the key to success in business, professional life, and personal
- Who will build bridges in the 21st century?
- If current trends continue, the answer will be Asians and Europeans.
They still believe in knowledge transmission and the critical importance of
specific, remembered knowledge. They still stand in awe of the amazing
knowledge-storing power of the human brain, and they're leaving us in the
academic dust, even though they typically educate 40+ students in their
- Of course you've heard about the enormous pressure placed upon Asian
kids, with many close to suicide. Of course it isn't true. It's just part
of the defensive strategy from the propaganda machine of the American
education establishment - a 600 billion dollar (annual) industry.
Copyright 1997-2011 William
G. Quirk, Ph.D.