- The Role of the NSF
- In 2008, The National Math Panel Identified the Foundations of Algebra

- The Foundations of Algebra - A Compact List

- What is NCTM Reform Math?
- The Reform Math Constructivist Teaching Philosophy
- The Key Fallacy Behind Reform Math
- Who is Bill Quirk?
- Essays by Bill Quirk

Beyond funding the development of these 13 NCTM reform math programs, the NSF has spent untold millions to support the nationwide adoption of these programs.

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.

- 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 page 46]
- Proficiency with whole numbers
is a
necessary precursor for the study of fractions. [NMP PDF
page 17]

- Computational
proficiency with
whole number operations is
dependent on sufficient and appropriate practice to develop automatic
recall of addition and related subtraction facts, and of
multiplication
and related division facts. It also requires fluency with the
standard
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
students should
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
pages 17
and 50]

- 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
and 78]

- 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
able
to analyze the properties of two- and three-dimensional shapes using
formulas to determine perimeter, area, volume, and surface area. [NMP
PDF page 46]

- Memorization
(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 arithmetic.

- Understanding 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.
- Understanding decimal place value number representation and knowing how to convert between fraction and decimal number representations.
- Understanding 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.

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 math programs also promote the ongoing use of hands-on "manipulatives." They say concrete tools must always be available and regularly used. They 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.

For one example of a K-5 NCTM reform math program, see TERC Hands-On Math: A Snapshot View

We are opposed to the NCTM version of "math reform." 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 how to add, subtract, multiply, and divide whole numbers, decimals, and fractions. These must be general skills, not limited to small, special case numbers.

- 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 math knowledge that all children should learn.
- Belief that knowledge should be naturally acquired as a byproduct of social interaction in real-world settings.
- Devaluation of teacher-centered knowledge transmission and learning from books.
- 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 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.

On April 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.”

Unfortunately, NCTM reform math is the “math wars” winner. The foundations of algebra topics are now missing from many American K-6 classrooms. Difficult to believe? Consider the writings of Marilyn Burns, the math program director for the Phil Mickelson ExxonMobil Teachers Academy. In her book,

- Click on The
Reform Math
Mindset Behind Phil
Mickelson ExxonMobil Math for more about Marilyn Burns view of standard pencil-and paper arithmetic.

Traditionally, K-6 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-6 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 lives in Boynton Beach FL and Guilford, CT.

- Review of Math Wars: A Guide for Parents and Teachers by Carmen Latterell

- Math Education Game-Changer: Khan Academy Math
- Common Core and Constructivist Math: Khan Academy Math Will Save the Day
- Standard Algorithm for Multiplication vs. Partial Products
- The Case for National Math Standards
- The Parrot Math Attack on Memorization
- The Bogus Research in Kamii and Dominick's Harmful Effects of Algorithms Papers
- 2008 TERC Math vs. 2008 National Math Panel Recommendations
- 2008 TERC Math vs. 2008 NMP Math: A Snapshot View
- The Reform Math Mindset Behind Phil Mickelson ExxonMobil Math
- A
Review of the
Voyages K-5 Math Program

- The Anti-Content Mindset: The Root Cause of the "Math Wars"
- Memorize Multiplication Facts? Cheney - Yes, Romberg - Abstained
- Understanding
the
Revised
NCTM Standards: Arithmetic is Still Missing

- Understanding
the
Original
NCTM Standards: They're Not Genuine Math Standards

- Chapter 1: Traditional K-12 Math Education
- Chapter 2: The NCTM Calls Them "Standards"
- Chapter 3: The NCTM Calls it "Math"
- Chapter 4: The NCTM Calls it "Learning Math"
- TERC Hands-On Math: A Snapshot View
- TERC Hands-On Math: The Truth is in the Details
- How the NCEE Redefines K-12 Math
- Chapter 1: A Summary View of NCEE Math
- Chapter 2:
- Chapter 3:
- Chapter 4:
- Searching For the Truth About the TIMSS 4th Grade Math Test
- Understanding
the
New
Jersey Math Standards: Bloated NCTM Standards

- Math Wars in Massachusetts
- Wall Street Journal - Numbers Racketl
- Guilford CT Public School Essays