The Truth About Constructivist Math
a graduate of Dartmouth
and holds a Ph.D. in Mathematics from The New Mexico State University.
Over a span of 8 years, he taught 26 different courses in math and
science at Penn State, Northern Illinois University, and Jacksonville
For a 15 year period,
Bill developed and presented courses dealing with interactive systems
His company, William G. Quirk Seminars, specialized in
and served hundreds of organizations, including AT&T,
America, FDIC, Federal Reserve Board, General Electric, General Foods,
Harvard Business School, Hewlett-Packard, Hughes Aircraft, IBM, MIT,
Oil, NASA, NIH, Texas Instruments, The Travelers, and The
Executive Office of the President of the United States.
Beginning in 1996, Bill
embarked on a
service endeavor to help parents besieged with constructivist math programs.
He is a major contributor to Mathematically Correct and a national advisor to NYC
HOLD and a co-author of The
State of State Math Standards 2005
Bill Quirk lives in Boynton
Beach FL and Stonington CT.
Recent Essays by Bill Quirk
The National Council of
Mathematics (NCTM) promotes constructivist math, because it believes
that traditional K-12 math is too difficult for most
also believe that traditional K-12 math is largely obsolete due to the
power of "technology." Accordingly, the NCTM promotes "math
standards" that substitute "math appreciation" content for
traditional K-12 math content. Their version of "math reform"
omits the essence of traditional K-12 math, including standard
computational skills, symbolic
manipulation skills, and mathematical reasoning skills. The
fills the void with calculator
"skills" and endless
busywork with hands-on "manipulatives."
We are opposed to the
reform." We know that math is a
domain, with standard arithmetic as the foundation. We know
can be all
over by the end of the sixth grade, if a child hasn't mastered the
and skills of standard pencil-and-paper arithmetic. By the
the 6th grade, students must understand carrying and borrowing, long
division, and how to compute with fractions. These must be
general skills, not limited to small, special case numbers.
all starts with memorization of the single-digit number facts for
addition, subtraction, multiplication, and division. By ingenious design,
computation is reduced to a set of these single-digit
facts. Without instant recall of the single digit
facts, the student's conscious mind will be bogged down figuring out
the specific single-digit facts needed to carry out each multidigit
computation. Next, without prior mastery of
computation, the student can't learn to compute with
fractions. But mastery of operations with fractions
gateway to algebra, and algebra is the gateway to higher
the math education of all students. We know that the poor
most from NCTM math, because there's no supplemental input from tutors
or well educated parents. We know that
majority of American children can learn genuine K-12 math.
fact is clearly proven by the fact that the vast majority of Asian
children do learn genuine K-12 math. Asian nations don't use
NCTM's approach. For a sample of what they do use,
NCTM mean by Math Reform?
Council of Teachers of
(NCTM) equates "math reform" with the ideas found in Principles and Standards
for School Mathematics (PSSM),
a 402 page
of the NCTM Standards. The NCTM calls it
math. Opponents call it "fuzzy" math or "new-new
of the name, "reform math" is characterized by an endorsement of
"constructivist" teaching methods and a rejection of
the content and skills of traditional
The NCTM has redefined the
math education. They believe that $5 calculators now cover
of arithmetic, graphing calculators now cover most of algebra, and
now cover most of the remainder of K-12 math. The
believe that traditional K-12 math only serves the needs of "the
elite," and they know that most K-12 math teachers are poorly prepared
to teach tradtional K-12 math. Putting it all together, the
and social goals, not traditional K-12 math. They promote
minimal learning expectations, with the constant use of calculators and
The NCTM is also excited
teaching methods. This philosophy is associated with the following beliefs:
- Belief that children must be
their own interests to personally discover the math knowledge that they
find interesting and relevant to their own lives.
Belief that knowledge should be
acquired as a byproduct of social interaction in real-world settings.
- Rejection of the concept of a
of basic math knowledge that all children should learn.
- Rejection of the traditional
education whereby teachers ask questions and present problems which
been carefully chosen to lead students to discover teacher-targeted
- Emphasis on peer knowledge, not
Belief in the primary importance
content-independent "process" skills.
- Devaluation of teacher-centered
- Emphasis on knowledge that is
Belief that learning must always
happy experience, with knowledge emerging naturally from games and
- Rejection of the need to
facts and skills of genuine mathematics.
- Rejection of the need for
- Rejection of the need to
to work harder.
Behind Constructivist Math
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 list."
Traditionally, K-12 math is
knowledge domain where American children build a remembered knowledge
of domain-specific content, with each child gradually coming to
hundreds of specific ideas that have been developed and organized by
contributors over thousands of years. With teachers who know math and
methods of knowledge transmission, the student is led, step-by-step, to
more and more specific math facts and skills, continually moving deeper
and deeper into the
knowledge domain that comprises traditional K-12 math. This
disciplined knowledge-building experience is a key enabler, developing
the memorizing and organizing skills of the mind, and thereby helping
prepare the individual to eventually build remembered knowledge bases
to other knowledge domains in the professions, business, or personal
The ongoing strength of our
economy depends fundamentally on a ready supply of millions of
workers who can learn to understand and extend thousands of specific
domains, from aeronautical engineering and carpentry to piano tuning
zoology. Although the specific facts, skills, and organizing
differ from domain to domain, genuine domain experts must necessarily
a vast amount of specific information that is narrowly relevant to
knowledge domains, frequently without the possibility of transfer to
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