The NCTM Calls it "Math"

Chapter 3 of Understanding the Original NCTM Standards by Bill Quirk

The NCTM Says Traditional K-12 Math is Obsolete

The "Introduction" to the NCTM Standards claims that the 54 "standards" details what mathematics students need to know". Traditionalists will be puzzled by this claim. They won't find "focused, specific, basic, teachable, and measurable" math content for the K-12 years, and they won't find traditional K-12 math. They'll find the NCTM has redefined the meaning of "math", frequently emphasizing that traditional K-12 math content should receive "decreased attention". Hard as it is to believe, they want to eliminate traditional K-12 math content and substitute calculator skills, math appreciation, and a whole range of general, content-independent skills. They want to emphasize social goals and psychological considerations, not traditional math content.

The NCTM wants you to believe that K-12 math should no longer cover the content that has been traditionally taught under the headings "arithmetic", "algebra", and "geometry". In the NCTM Standards: Introduction we are told:

The Truth:

  1. Most of math isn't impacted by calculators and computers.
  2. Math is a constantly growing field, but the foundations of math haven't changed.
  3. Kids still need to learn the traditional content of "arithmetic", "algebra", and "geometry". With the NCTM's approach they'll be limited to calculator skills. They'll never be able to learn more advanced math, they'll be denied access to other knowledge domains that depend on math, and they'll never acquire the logical and abstract thinking skills that only learning math conveys.
If you buy "traditional K-12 math is obsolete", the NCTM has you set up to accept their strategy for replacing traditional K-12 math content:

The Anti-Content, Anti-Memory Progressive Mindset

The NCTM Standards have been developed by progressive "math educators", not by people with genuine knowledge of mathematics. For eighty years progressive educationists have rejected the idea of remembering any domain-specific knowledge.  They say knowledge is changing too fast, and the facts of today will be obsolete tomorrow. Calculators and computers are the latest "proof" of this claim. The NCTM wields them as a double-edged sword, justifying the trashing of traditional math and offering the benefit of exciting "tools" for bypassing the difficulties of traditional "paper-and-pencil" math.

Progressive educationists believe important factual knowledge is already known intuitively or will be picked up naturally as a byproduct of real-world experiences. They claim that real-world experts rely on "higher-order skills" and "just-in-time" factual knowledge supplied by computers and reference materials. They say real-world experts never trust their own long-term memory.

The Truth:

  1. Real-world knowledge experts can't afford the time for constantly looking up information.
  2. Expert knowledge is remembered from thousands of sources and experiences. It isn't easily available in convenient reference materials.
  3. "Just-in-time knowledge" is a total mischaracterization of how all knowledge experts work. Their value as experts depends fundamentally on the depth and breadth of their remembered domain-specific knowledge. Sure, they use computers and reference materials, but not constantly. Too much is expected too fast.
  4. What fools would belittle and deny the unbelievable power of human memory?
In his recent book, The Schools We Need & Why We Don't Have Them, Professor E. D. Hirsch, Jr. explains the historical process that led to the current unbelievable mindset of today's K-12 education establishment:
  1. Historically, American teacher-training institutions were primarily concerned with teacher mastery of subject matter. Content was primary and teaching methods were secondary.
  2. Teacher-training institutions lost responsibility for content as they were absorbed into universities.  For example, the math department would now teach math..
  3. The educationists were left with methods, so they made methods primary and began to disparage the importance of remembered content.
  4. Beginning eighty years ago, the anti-content ideas of William Heard Kilpatrick began to dominate American public education.
  5. American schools of education now preach content-independent methods and general skills. They reject traditional knowledge transmission and the building of a remembered knowledge base of domain-specific facts and skills.
The result? Today's graduates of American schools of education have minimal knowledge of content and they deny the power of human memory. This is especially true for K-8 teachers. More importantly, they have been indoctrinated to believe that remembering content isn't important.


The NCTM Says Math Appreciation is the First Goal

The Truth:

  1. The NCTM is talking about history and sociology, not math. They are offering a convenient distraction for those who want to avoid the difficulties of teaching genuine math.
  2. The first goal should be learning "K-12 math subset", the specific math content that should be taught and learned during the K-12 years. Ideally, the specifics would be identified by national math standards that meet the characteristics of quality listed in chapter 2.
  3. The NCTM never explains why appreciating math is more important than learning math.
  4. The NCTM Standardscontain no examples of the "cultural, historical, and scientific evolution of mathematics" Fifty-eight documents and 258 pages, but no illustrations of the primary goal.

The NCTM Says Traditional K-12 math content should be replaced by:

Calculators and Computers, Not Paper-and-Pencil

The Truth:

  1. Although the first quote above suggests that the NCTM expects kids to remember basic addition and multiplication facts, this is never explicitly stated in the NCTM Standards. More generally, the NCTM rejects the idea of remembering specific math knowledge. The "availability of calculators" isn't the problem. The problem is the attitude of the NCTM.
  2. Calculators should not be introduced before the student can instantly recall basic math facts and has completely mastered the paper-and-pencil skills of traditional arithmetic. Without a calculator:
  3. If calculator and computer skills are substituted for traditional K-12 math skills, students will never be able to build more sophisticated math knowledge. They will be forever limited to what can be done with calculators and computers.

General Problem-Solving Skills, Not Specific Math Knowledge

The Truth:

  1. The NCTM has redefined the meaning of "mathematical problem solving". They don't believe in first learning math and then using it to solve problems. They see "trail and error" and other content-independent "problem solving" skills as their way of applying the fundamental progressive gospel of "discovery learning" to K-12 math..
  2. "Doing math" is not synonymous with these general content-independent skills.
  3. Good teachers pose problems as a way to introduce new math topics, but the problems are carefully chosen to lead the students to the desired learning experience.

General Communication Skills, Not the Language and Symbols of Math

The Truth:

  1. Yes! General communication skills are of fundamental importance. They're more important than math skills. But they're not math. Math time should be reserved for learning math.
  2. The "technical vocabulary and symbolism" of math has evolved over centuries. The precise language and symbols of math provide a powerful universal vehicle for clear and concise communication. It is simply astounding that "math educators" can suggest it is possible to "know math" without knowing the vocabulary and symbols of math.

The NCTM Says Precision and Exactness Should be Replaced by:

An Attitude Adjustment About "Correct Answers"

The Truth:

  1. Certainly there can be more than one mathematically correct way to get the answer, but do we want bridges built by kids who believe 9 times 7 is 97? Should we be satisfied if Johnny, Sam, and Sarah are each happy with their own answer, even if none of them agree?
  2. The NCTM sees math through the subjective prism of progressive social science. Just as they claim to honor differences of opinion, they want to honor every kid's answer as valuable and praiseworthy. But math isn't subjective. Precision and exactness are its principle hallmarks.

Intuitive Arithmetic, not Memorizing Addition and Multiplication Facts

The Truth:

  1. Progressive educators believe that the only important knowledge is already known intuitively, somehow hidden in the brain. See Discovery Learning in Chapter 4.
  2. Arithmetic isn't natural or intuitive. It's a completely man-made abstraction. You have to be told that 1 + 1 = 2. You're not born with this knowledge, and you'll never pick it off a tree.
  3. There isn't time to "figure it out each time". What's the point? Why not use the immense power of human memory?
  4. Calculators may get you through everyday life, but if you never commit to memory such basic facts as the 9 by 9 addition and multiplication tables, the resulting memory gaps will prevent you from ever building a coherent math knowledge base in your brain.

Informal and Inductive Reasoning , not Deductive Reasoning

Note: "Informal" reasoning refers to the use of concrete materials (manipulatives).
"Inductive" reasoning refers to generalizing from multiple observations.

The Truth:

  1. As the first three quotes indicate, the NCTM endorses "informal" proofs (the use of concrete "manipulatives") for all K-12 years. Then, in the fourth quote, they admit to teaching something that isn't true. Amazing!
  2. Informal reasoning is not math. (See manipulatives in chapter 1). Prolonged reliance on concrete "pacifiers" interferes with the most important social reason for studying math, the development of the average citizen's ability to think abstractly.
  3. Inductive reasoning is the method of science, not math. Rather than being used "often" in math, it plays a very minor role and doesn't qualify as teachable math.
  4. If kids are to learn about "reasoning" under the heading "math", it should be deductive reasoning.

Estimation as an Important Part of Imprecise Mathematics

The Truth:

  1. The NCTM has necessarily elevated a bit player to a starring role. The elevation of estimation is a logical consequence of the glorification of calculator skills.
  2. The emphasis on estimation is another time-consuming distraction.
  3. Why can't the student determine exactly how long it will take to buy the bike?

Mathematics is the Science of Pattern Recognition

The Truth:

  1. Pattern recognition, like inductive reasoning, is more properly associated with science, not math. Here it is one more distraction away from genuine math.
  2. Recognizing patterns depends on remembered math facts. Recognizing the pattern in the sequence (5, 10, 17, 26, 37, 50, ...) requires a knowledge of perfect squares. Recognizing the pattern in (10, 12, 16, 18, 22, 28..) requires a knowledge of prime numbers.

The NCTM Still Calls it "Algebra" and "Geometry"

The Truth:

  1. The NCTM wants to substitute calculator skills for the logical and abstract thinking skills that can only be learned through the mathematical disciplines of algebra and plane geometry.
  2. The NCTM is saying that kids shouldn't have to remember specific math terminology, math facts, or math skills. Somehow, in the future, they'll recognize what they need to know, and they'll know how to "look it up".

Next?

Finding the Context for a Quote from the NCTM  Standards

Update:  The NCTM no longer supports internet access to the (original) NCTM Standards.  You now need the hardcopy.  You can no longer search and "Find" as indicated below.

The NCTM Standards begins with an Introduction document and then presents fifty-four standards documents, divided into four sections:  Grades K-4, Grades 5-8, Grades 9-12, and Evaluation. Each section is preceded by a section overview document.

Chapters 3 and 4 use quotes from the NCTM Standards. Each quote is identified by a code of the form S.D, where "S" represents the section identifier, and "D" usually identifies the standard number, with the exception of D value of "O" identifying the section overview document. The section identifiers codes are K-4, 5-8, 9-12, and EVAL. For example, K-4.5 indicates the fifth standard of the K-4 section, and 9-12.O indicates the overview to the 9-12 section.. Finally, "Intro" indicates the Introduction to the entire collection of NCTM Standards The S.D code allows you to find the source document. Then, using the Find command, you can see the quote in context.


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