The Truth About Constructivist Math

By Bill Quirk [wgquirk@wgquirk.com]

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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.

What Does the NCTM mean by Math Reform?

The National Council of Teachers of Mathematics (NCTM) equates "math reform" with the ideas 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.  This philosophy is associated with the following beliefs:

  1. 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.
  2. Belief that knowledge should be naturally acquired as a byproduct of social interaction in real-world settings.
  3. Belief in the primary importance of general, content-independent "process" skills.
  4. Belief that learning must always be an enjoyable, happy experience, with knowledge emerging naturally from games and group activities.

The Key Fallacy Behind 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.


Essays By Bill Quirk


Who is Bill Quirk? (wgquirk@wgquirk.com)

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.


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Copyright 1997 - 2014 William G. Quirk, Ph.D.