The Truth About Constructivist Math
New in 2014
New in 2013
About
This
Site
The National Council of
Teachers
of
Mathematics (NCTM) promotes constructivist math, because it believes
that traditional K-12 math is too difficult for most
children. They
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
NCTM
fills the void with calculator
"skills" and endless
busywork with hands-on "manipulatives."
We are opposed to the
NCTM version
of
"math
reform." We know that math is a
vertically-structured
knowledge
domain, with standard arithmetic as the foundation. We know
it
can be all
over by the end of the sixth grade, if a child hasn't mastered the
facts
and skills of standard pencil-and-paper arithmetic. By the
end of
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.
It
all starts with memorization of the single-digit number facts for
addition, subtraction, multiplication, and division. By ingenious design,
each
multidigit
computation is reduced to a set of these single-digit
facts. Without instant recall of the single digit
number
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
multidigit
computation, the student can't learn to compute with
fractions. But mastery of operations with fractions
is the
gateway to algebra, and algebra is the gateway to higher
mathematics.
We are
concerned
about
the math education of all students. We know that the poor
suffer
most from NCTM math, because there's no supplemental input from tutors
or well educated parents. We know that
the vast
majority of American children can learn genuine K-12 math.
This
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
the
NCTM's approach. For a sample of what they do use,
visit SingaporeMath.com.
K-12 Math Education
Essays By Bill Quirk
What
Does the
NCTM mean by Math Reform?
The
National
Council of Teachers of
Mathematics
(NCTM) equates "math reform" with the ideas currently found in Principles and Standards
for School Mathematics (PSSM),
a 402 page
revision
of the NCTM Standards. The NCTM calls it
"standards-based"
math. Opponents call it "fuzzy" math or "new-new
math."
Regardless
of the name, "reform math" is characterized by an endorsement of
"constructivist" teaching methods and a rejection of
the content and skills of traditional
K-12 math.
The NCTM has redefined the
goals
of K-12
math education. They believe that $5 calculators now cover
most
of arithmetic, graphing calculators now cover most of algebra, and
computers
now cover most of the remainder of K-12 math. The
NCTM also
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
NCTM
emphasizes math
appreciation
and social goals, not traditional K-12 math. They promote
minimal learning expectations, with the constant use of calculators and
hands-on manipulatives.
The NCTM is also excited
about
"constructivist"
teaching methods. Purists will argue about the meaning of
this
term,
but this philosophy is associated with the following beliefs:
- Belief that children must be
allowed
to follow
their own interests to personally discover the math knowledge that they
find interesting and relevant to their own lives.
- Rejection of the concept of a
common
core
of basic math knowledge that all children should learn.
- Rejection of the traditional
process
of math
education whereby teachers ask questions and present problems which
have
been carefully chosen to lead students to discover teacher-targeted
math
knowledge.
- Emphasis on peer knowledge, not
teacher knowledge.
- Belief that knowledge should be
naturally
acquired as a byproduct of social interaction in real-world settings.
- Devaluation of teacher-centered
classroom
learning
and learning
from books.
- Emphasis on knowledge that is
needed
for everyday
living.
- Belief in the primary importance
of
general,
content-independent "process" skills.
- Rejection of the need to
remember
the specific
facts and skills of genuine mathematics.
- Belief that learning must always
be an
enjoyable,
happy experience, with knowledge emerging naturally from games and
group
activities.
- Rejection of the need for
memorization and
practice.
- Rejection of the need to
challenge a
child
to work harder.
The
Key Fallacy
Behind Fuzzy
Math (Constructivist Math)
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 list."
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.
Bill
Quirk is
a graduate of Dartmouth
College
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
computer
science at Penn State, Northern Illinois University, and Jacksonville
University.
For a 15 year period,
beginning in
1981,
Bill developed and presented courses dealing with interactive systems
design.
His company, William G. Quirk Seminars, specialized in
software
usability
and served hundreds of organizations, including AT&T,
Bank of
America, FDIC, Federal Reserve Board, General Electric, General Foods,
Harvard Business School, Hewlett-Packard, Hughes Aircraft, IBM, MIT,
Mobil
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
public
service endeavor to help parents besieged with new "math"
programs.
He is a major contributor to Mathematically
Correct, 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.
Recommended
Sites
Recommended
Books
- Knowing and Teaching Elementary
Mathematics
by Liping Ma
- Teacher' Understanding of
Fundamental Mathematics
in China and the United States
- The Schools We Need & Why
We
Don't
Have Them by E.D. Hirsch, Jr.
- Dumbing Down Our Kids by
Charles J.
Sykes
- The Learning Gap by Harold
W.
Stevenson
and James W. Stigler
- Begin Here by Jacques Barzun
- Connected Knowledge by Alan
Cromer
Recommended
Net
Documents
About
Copyright
You
may print
and distribute essays by
Bill
Quirk. You may sell copies to recover printing costs, but not
for
profit.
Copyright
1997 - 2013 William G. Quirk, Ph.D.