The NCTM Reform Math Deception
The Foundations of Algebra Are Missing!
It's All Been Funded by the NSF!
Links to Sections of this Essay:
The Role of the NSF
Since the original NCTM Standards
were released in 1989, the National Science Foundation (NSF) has spent
over 100 million dollars to promote the math "content" and
constructivist math teaching philosophy promoted by the NCTM. In
particular, the NSF has funded the development of 3"exemplary"
elementary math programs, 5 "exemplary" middle school math programs,
and 5 "exemplary" high school math programs. A program is
considered "exemplary" if it is "consistent with the NCTM
Standards". Click here for a complete list of these 13 NCTM reform math programs.
Beyond funding the development of these 13 NCTM reform math programs,
the NSF has spent untold millions to support the nationwide adoption of
There are 2 major problems with these NSF funded NCTM reform math programs:
- NCTM reform math programs
reject traditional teacher-centered direct instruction. Instead,
they promote student-centered constructivist (discovery) learning.
- NCTM reform math programs omit the foundations of algebra.
- But algebra is the gateway to higher math. So "graduates" of these programs have no math future. Difficult to believe, but true.
In 2008, the National Math Panel Identified the Foundations of Algebra
The 2008 Final Report of the National Math Panel (NMP PDF) makes several observations
and/or recommendations about how students need to be prepared for
school algebra. Here are key "foundations of algebra" quotes from
this report. [Bold and underline emphasis added].
- Proficiency with whole numbers,
fractions, and particular aspects of geometry and measurement should be
understood as the Critical Foundations of Algebra. Emphasis on these
essential concepts and skills must be provided at the elementary and
middle grade levels. [NMP PDF page 46]
- The coherence and sequential
nature of mathematics dictate the
foundational skills that are necessary for the learning of algebra. The
most important foundational skill not presently developed appears to be
proficiency with fractions (including decimals, percents, and negative
fractions). The teaching of fractions must be acknowledged as
critically important and improved before an increase in student
achievement in algebra can be expected. [NMP PDF
- Proficiency with whole numbers
necessary precursor for the study of fractions. [NMP PDF
whole number operations is
dependent on sufficient and appropriate practice to develop automatic
recall of addition and related subtraction facts, and of
and related division facts. It also requires fluency with the
algorithms for addition, subtraction, multiplication, and division.
Additionally it requires a solid
understanding of core concepts, such as the commutative, distributive,
and associative properties. [NMP PDF page 19]
- By the term
proficiency, the Panel means that
understand key concepts, achieve automaticity as appropriate (e.g.,
with addition and related subtraction facts), develop flexible,
accurate, and automatic
execution of the standard algorithms, and use
these competencies to solve problems. [NMP PDF
- The Panel cautions
that to the degree that
calculators impede the development of automaticity, fluency in
computation will be adversely affected. [NMP PDF pages 24
- Difficulty with
fractions (including decimals and percents) is pervasive and is a major
obstacle to further progress in mathematics, including algebra. A
nationally representative sample of teachers of Algebra I who were
surveyed for the Panel rated students as having very poor preparation
in “rational numbers and operations involving fractions and decimals.”
[PDF page 19]
- Before they begin algebra
course work, middle school students
should have a thorough understanding of positive as well as negative
fractions. They should be able to locate positive and negative
fractions on a number line; represent and compare fractions, decimals,
and related percents; and estimate their size. They need to know that
sums, differences, products, and quotients (with nonzero denominators)
of fractions are fractions, and they need to be able to carry out these
operations confidently and efficiently. They should understand why and
how (finite) decimal numbers are fractions and know the meaning of
percentages. They should encounter fractions in problems in the many
contexts in which they arise naturally, for example, to describe rates,
proportionality, and probability. Beyond computational facility with
specific numbers, the subject of fractions, when properly taught,
introduces students to the use of symbolic notation and the concept of
generality, both being integral parts of algebra. [NMP
PDF page 46]
- Furthermore, students should be
to analyze the properties of two- and three-dimensional shapes using
formulas to determine perimeter, area, volume, and surface area. [NMP
PDF page 46]
The Foundations of Algebra - a Compact List
(automatic recall) of single digit addition facts, subtraction
facts, multiplication facts, and division facts.
- This is the
key necessary condition for later mastery of the standard
algorithms for multi-digit computation. For example, the
second grader should instantly know that 7 + 8 = 15, and the fourth
grader should instantly know that 7 x 8 = 56. Students who
don't instantly know single digit number facts will get bogged down
when they encounter the standard algorithms for paper-and-pencil
both compact and expanded place value number representation, including the concepts of
carrying for addition and borrowing for subtraction.
- Know that that the expanded place value representation of 467 is 4 x 100 + 6 x 10 + 7 and this is the definition of 467.
- Know that carrying and borrowing is best understand by
showing the computation with the numbers first shown in compact place
value form and then showing the numbers in expanded place value form.
- Mastery of the
standard algorithms for mult-digit addition, subtraction, multiplication and
division, first for whole numbers and later for decimals.
- Appreciate the
ingenious design by which each multi-digit computation is reduced
to a set of single digit facts.
- Appreciate the
ingenious design by which a problem in multi-digit multiplication is reduced to
a problem in multi-digit addition.
- Appreciate the
ingenious design by which a problem in multi-digit (long) division is reduced to
problems in multi-digit multiplication and multi-digit subtraction.
- There's great
power in the fact that carrying and borrowing work the same way for
whole numbers and decimals, and there's great power in the ability to automatically
recall the single digit number facts.
- Mastery of the
standard procedures for addition, subtraction, multiplication, and
division of fractions. This includes understanding the concept of equivalent
fractions and how to find a common denominator.
decimal place value number representation and knowing how to convert
between fraction and decimal number representations.
how fractions and decimals are represented on the number line.
- Knowing how to
find the area and perimeter of triangles, rectangles, and circles.
- Knowing how to
find the volume and surface area of basic 3-dimensional shapes.
What is NCTM Reform Math?
National Council of Teachers of Mathematics (NCTM) released the NCTM
Standards in 1989. This document omits the foundations of algebra
described in the section above. The NCTM believes this content is
now obsolete, due to the power of "technology." Rather than
traditional pre-algebra content, the NCTM promotes the reform math
constructivist philosophy of
Because of the many millions spent by the NSF, NCTM
reform math now dominates elementary math education in K-6 public
schools. Reform math educators promote the ongoing use of
calculators, beginning in kindergarten. Reform
programs also promote the ongoing use of hands-on
say concrete tools must always be available and regularly used.
reject the idea that children must eventually migrate from hands-on to
abstract thinking. Reform math educators believe
that K-6 math education should be limited to their concept of
everyday math needs. They fail to appreciated the vertically-structure
of the math knowledge domain. They’re blind to the fact that students
can’t learn algebra, if they haven’t first mastered the foundations of
algebra. They’re blind to the fact that algebra is the
gateway to higher learning in the STEM fields, where STEM denotes,
Science, Technology, Engineering, and Mathematics.
are opposed to the
reform." 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 how to add, subtract, multiply,
and divide whole numbers, decimals, and fractions. These must be
general skills, not limited to small, special case numbers.
Reform Math Constructivist Teaching Philosophy:
- Belief that children must be
their own interests to personally discover the math knowledge
find interesting and relevant to their own lives.
Belief that knowledge should be
acquired as a byproduct of social interaction in real-world
- Rejection of the concept of a
of math knowledge that all children should learn.
Belief in the primary importance
content-independent "process" skills.
- Devaluation of teacher-centered
knowledge transmission and learning
- Promotion of
student-centered discovery learning, believing that students can
effectively learn the math they need to learn as a byproduct of
carrying out projects and investigations, listening primarily to their
peers, not the teacher.
- Emphasis on knowledge that is
Belief that learning must always
happy experience, with knowledge emerging naturally from games
- Rejection of the need to
facts and skills of genuine mathematics.
- Rejection of the need for
12, 2000, the NCTM released a revision of the NCTM Standards. The
next day The New York Times reported: "In an important about-face, the
nation's most influential group of mathematics teachers announced
yesterday that it was recommending, in essence, that arithmetic be put
back into mathematics, urging teachers to emphasize the fundamentals of
computation rather than focus on concepts and reasoning." But
this was all a sham. That same day the following contradictory
statement was posted at the NCTM website: “More than ever,
mathematics must include the mastery of concepts instead of mere
memorization and the following of procedures. More than ever, school
mathematics must include an understanding of how to use technology to
arrive meaningfully at solutions to problems instead of endless
attention to increasingly outdated computational tedium.”
NCTM reform math is the “math wars” winner. The
foundations of algebra topics are now missing from many American
K-6 classrooms. Difficult
believe? Consider the writings of Marilyn Burns, the math program
director for the Phil Mickelson ExxonMobil Teachers Academy. In her book, About
Teaching Mathematics, Marilyn
“because of the present availability of calculators, having
spend more than six years of their schooling mastering
paper-and-pencil arithmetic is as absurd as teaching them to ride and
care for a horse in case the family car breaks down.” Ms.
Burns claims that “the emphasis of arithmetic instruction should
be on having students invent their own ways to compute.”
But there’s no inventing. Children are taught
nonstandard methods for solving simple, special case problems. Parents
and grandparents can’t help, because it’s not the math
methods they use.
has reform math achieved such success? Major funding by the NSF is the key reason. Next, it’s not easy
to correctly teach the foundations of algebra, and American schools of
education typically offer just one 3-hour survey course in math.
So most elementary schools teachers are not prepared to
teach the foundations of algebra topics.
been an extensive propaganda campaign that has praised the reform
approach and trashed standard pencil-and-paper arithmetic.
The Key Fallacy Behind Reform Math
Reform [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
picture" or "make a list."
K-6 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,
more and more specific math facts and skills, continually moving
and deeper into the
knowledge domain that comprises traditional K-6 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
is Bill Quirk?
Quirk is a graduate of Dartmouth College and holds a Ph.D in
Mathematics from New Mexico State. 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,
Bill developed and presented courses dealing with interactive
His company, William G. Quirk Seminars, specialized in
and served hundreds of organizations, including AT&T,
America, FDIC, Federal Reserve Board, General Electric, General
Harvard Business School, Hewlett-Packard, Hughes Aircraft, IBM,
Oil, NASA, NIH, Texas Instruments, The Travelers, and The
Executive Office of the President of the United States.
in 1996, Bill
embarked on a
service endeavor to help parents besieged with reform constructivist math
He is a major contributor to Mathematically
Correct and a national advisor to NYC
HOLD and a co-author The State
of State Math Standards 2005
Bill Quirk lives in
Beach FL and Guilford, CT.
By Bill Quirk
and distribute essays by Bill Quirk. .
1997-2017 William G. Quirk, Ph.D.