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

by Bill Quirk (E-Mail: wgquirk@wgquirk.com)

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:

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.
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

"Students don't need to know multiplication tables in order to divide, they say. Using objects and calculators they can figure it out - and thus begin to create their own mathematical knowledge."
"According to the council, stressing addition, subtraction, and worst of all, memorization made students into passive receivers of rules and procedures rather than active participants in creating knowledge."
"The standards recommended that students get together with peers in cooperative learning groups to "construct" strategies for solving math problems, rather than sit in class with teachers instructing them."
"Calculators were a necessity from kindergarten on, the council said, because students liberated from "computational algorithms" could pursue higher-order activities, like inventing personal methods of long division."

Thomas Romberg Defended the NCTM

"Although the council wants all students to learn to estimate and use mental calculation and to be equally comfortable using paper and pencil, calculators and computers, nowhere does it argue that students do not need to memorize multiplication facts. Nor does the council say that students should use a calculator for all computations."
"Similarly, the examples the teachers council has produced often portray students working on real-world problems, providing oral or written explanations of how a problem was solved, or collaborating with other students. But the council never meant to imply that such approaches are appropriate for every lesson."
"Sometimes even appropriate material is ineffective because the teacher using it doesn't have enough math background or the right training"
"It's unfair to attack the entire program because of initial missteps and isolated examples of misapplied guidelines."

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:

First, and most importantly, the NCTM Standards discourage the memorization of multiplication facts and any other specific math content. Whenever memorization or memorization methods (practice) are mentioned, the tone is always negative. Here are a few examples:
- "The curriculum should focus on the development of understanding, not on the rote memorization of formula."
- "The 9-12 standards call for a shift in emphasis from a curriculum dominated by memorization of isolated facts and procedures and by proficiency with paper-and-pencil skills to one that emphasizes conceptual understandings, multiple representations and connections, mathematical modeling, and mathematical problem solving".
- "Classroom observations should gather information about whether mathematics is portrayed as an integrated body of logically related topics as opposed to a collection of arbitrary rules that students must memorize."
Next, Thomas Romberg is literally correct, the NCTM Standards don't say that "students should use a calculator for all computations". Here's what they do say:
- "Contrary to the fears of many, the availability of calculators and computers has expanded students' capability of performing calculations. There is no evidence to suggest that the availability of calculators makes students dependent on them for simple calculations. Students should be able to decide when they need to calculate and whether they require an exact or approximate answer. They should be able to select and use the most appropriate tool."
  - The clear message is that all kids, regardless of age, should decide for themselves about the appropriate use of calculators. (So far, there are no known instances of children who have personally decided to reject the calculator in favor of the more difficult option of learning traditional arithmetic.)
Finally, what do the NCTM Standards have to say about group learning?
- "This approach instills in students an understanding of the value of independent learning and judgment and discourages them from relying on an outside authority to tell them whether they are right or wrong."
- "Students should frequently work together in small groups to solve problems.... They should verify their own thinking rather than depend on the teacher to tell them whether they are right or wrong."

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:

New-new "problem-solving" doesn't depend on first learning specific math content.
New-new "communicating" doesn't emphasize the correct use of the precise symbols and language of math.
New-new "reasoning" doesn't refer to the step-by-step application of remembered math content.
New-new "connecting" isn't about relating new math knowledge to previously learned math.

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

The NCTM problem-solving strategies are:
1. "Using manipulative materials"
2. "Trial and error"
3. "Making a list"
4. "Drawing a diagram"
5. "Looking for a pattern"
6. "Acting out a problem"
7. "Guess and check"
Grade K-4 NCTM communication is about:
- "the integration of language arts as children write and discuss their experiences in mathematics."
- "children might draw pictures and then tell or write stories about the equation "
- "Students can write a letter to tell a friend about something they have learned in mathematics class."
Grade 9-12 teachers of NCTM communication are to:
- "Direct instruction away from a focus on the recall of terminology and routine manipulation of symbols and procedures."
- Recognize that "in mathematics, just as with a building, all students can develop an understanding and appreciation of its underlying structure independent of a knowledge of the corresponding technical vocabulary and symbolism."
- Understand that "technical symbolism should evolve as a natural extension and refinement of the students' own language".
NCTM reasoning is about:
- Helping "children learn that mathematics is not simply memorizing rules and procedures but that mathematics makes sense, is logical, and enjoyable".
- "Creating and extending patterns of manipulative materials and recognizing relationships within patterns".
- Recognizing that "most fifth graders are still concrete thinkers who depend on a physical or concrete context for perceiving regularities and relationships".
- Understanding that "even the most advanced students at the 5-8 level might use concrete materials to support their reasoning".
- Encouraging students "to explain their reasoning in their own words".
- Recognizing that, prior to grade 9, "experience both in and out of school has taught them to accept informal and empirical arguments as sufficient", (Note: "Informal arguments" refers to the use of concrete manipulatives, and "empirical arguments" refers to generalizing from observations.)
Under the heading of connections, the NCTM wants to:
- Insure that children "will not need to learn or memorize as many procedures; and will have the foundation to apply, recreate, and invent new ones when needed".
- Help students to "view mathematics as an integrated whole rather than as an isolated set of topics".
- Insure that "as in the earlier grades, teachers in grades 9-12 should introduce a new topic by exploring appropriate concrete representations".

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

Mathematicians reject the idea that you can "do math" without "knowing math".
- Genuine mathematicians know that a necessary condition for reasoning mathematically is a remembered knowledge base of specific math facts and specific math skills
Mathematicians reject the early use of calculators, beginning in kindergarten, and they question the current excessive use of calculators, during all of the K-12 years.
Mathematicians want parents to know that kids must be challenged, not babied until grade 9 and beyond with concrete "manipulatives", and not continually waiting until the child feels "ready".
Mathematicians reject the substitution of "informal and empirical arguments" for genuine mathematical reasoning and genuine mathematical proof.
Mathematicians cringe at the belief that math appreciation (a cult-like excitement about the "power" and "beauty" of math) is more important than actually learning specific math content.
Mathematicians reject the belief that general, content-independent skills are in any way central to the process of learning mathematics.

Anti-Content Thinking Threatens All Americans

If our kids never learn the importance of remembered knowledge, and if they are programmed to think that memorization and practice are not necessary, then what happens if they somehow reach medical school and need to quickly memorize thousands of facts from Gray's Anatomy? It's difficult enough even with the traditional preparation of the mind.
- This is not just about kids who go to medical school. The current reigning educational philosophy is dangerous for all our children. If they are to be successful in life, they must effectively use the amazing knowledge-storing power of their brains. Are we really going to continue to let today's educationists program our kids to believe that remembering specific knowledge is a bad idea, and that computer "tools" and "look-up skills" are the key to success in business, professional life, and personal knowledge-based interests?
Who will build bridges in the 21st century?
- If current trends continue, the answer will be Asians and Europeans. They still believe in knowledge transmission and the critical importance of specific, remembered knowledge. They still stand in awe of the amazing knowledge-storing power of the human brain, and they're leaving us in the academic dust, even though they typically educate 40+ students in their classrooms.
  - Of course you've heard about the enormous pressure placed upon Asian kids, with many close to suicide. Of course it isn't true. It's just part of the defensive strategy from the propaganda machine of the American education establishment - a 600 billion dollar (annual) industry.

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