Memorize Multiplication Facts? 
Cheney, Yes.    Romberg, Abstain.

by Bill Quirk (E-Mail:

This Essay Deals with a Major Fallacy of NCTM Math:  Memorization and Practice are Ruled Out.  So Kids Don't Remember Specific Math Facts and Skills.  But "Doing Math" Requires the Application of Remembered Math Facts and Skills That Must First be Stored in the Brain. 

The First Fallacy of NCTM Math is the Lack of Genuine Math Content.

2005 Update Notes
  1. Originally written in 1997, this essay refers to the 1989 version of the NCTM Standards.  But it's still relevant, because the 2000 version of the NCTM Standards is based on the same constructivist philosophy. 
  2. NCTM math is synonymous with "new-new math," "fuzzy math," and "constructivist math."

On August 11, 1997 the New York Times published two Op-Ed pieces, one by Lynne Cheney, former chair of the National Endowment for the Humanities, and the other by Thomas Romberg, former chair of the Commission on Standards of The National Council of Teachers of Mathematics (NCTM).

To view the Cheney and Romberg essays, click on the author's name below.

Lynne Cheney Described the "Constructivist" Philosophy of the NCTM

 Thomas Romberg Defended the NCTM

What Do the NCTM Standards Actually Say?

Thomas Romberg suggests that the NCTM Standards have been misunderstood. They don't explicitly rule out all memorization, they don't recommend the constant use of calculators, and they don't say kids must always work in groups, attempting to solve real-world problems. We are to be comforted that the NCTM  never intended an extreme interpretation of their "less of this" and "more of that" recommendations. They never said that "none" was the optimal form of "less", or that "always" was the optimal form of "more". Apparently it's all the fault of poorly prepared teachers. Of course, the answer is increased funding for NCTM-controlled teacher training.

Here's what the NCTM Standards actually say:

A Major Fallacy Behind NCTM Math

The NCTM's 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.

If Traditional Content is Out, What's NCTM Math "Content"? 

The major subtopics are calculator skills, math appreciation, and, general, content-independent "process" skills. For example, in his New York Times Op-Ed piece (August 11, 1997), Thomas Romberg placed the spotlight on general, content-independent skills when he wrote about the foundational role of the "four general standards - problem solving, communication, reasoning, and connection". Of course, space didn't permit him to give you the new-new definitions of these high-sounding terms. Fortunately, we have space here. The truth is in the details.

First, forget your old-old ideas:

Quoting from the NCTM Standards, here's the new-new way:

Brought To You by "Math Educators", Not Mathematicians

When Thomas Romberg  tells you that the NCTM Standards have "received wide support from mathematicians and math educators", he's hoping you're not too picky about the definition of "mathematician". Sure, there are a few mathematicians who appear to go along, but these typically haven't read the NCTM Standards and just assume that they can't be that bad. (Of course, "a few" suffices as a "proof" for Thomas Romberg, using the new-new math definition of "empirical reasoning".)

Math Educators Do Not Speak for Mathematicians

Anti-Content Thinking Threatens All Americans

Copyright 1997-2011 William G. Quirk, Ph.D.