The NCTM Calls it "Learning Math"

Chapter 4 of Understanding the Original NCTM Standards  By Bill Quirk


The NCTM Says Standards Promote Social Goals

The Truth:

  1. None of these goals deals with genuine math. The NCTM is interested in sociology, psychology, and general content-independent skills, not genuine K-12 math content. Don't be confused by the NCTM's use of the terms "mathematics" and "mathematically". They're referring to their redefinition of "math", as described in chapter 3.
  2. K-12 math standards should clearly identify specific academic content for each K-12 year.

Building Self-Confidence is More Important Than Learning Math

The Truth:

  1. This is the second goal, after math appreciation. The NCTM is saying self-esteem is more important than learning math. They never says how this goal can be achieved, and they never suggest that learning math will help to improve self-confidence. For the NCTM, self-confidence is the goal, not learning genuine math.
  2. This is fundamental progressive gospel, with "self-esteem" now called "self-confidence".
  3. Math standards shouldn't deal with social and psychological considerations.

"Developmentally Appropriate" Learning Takes Considerable Time

The Truth:

  1. Why hasn't Johnny learned arithmetic by the eighth grade? The answer? The NCTM has achieved their goal if Johnny appears to be "happy". Learning math is not required. Kids must be free to develop at their own "natural pace". They're fragile and can't be rushed. Current "happiness" is the goal. There's no concern about what will happen to them after the K-12 years.
  2. Recently, genuine research in cognition has disproved "developmentally appropriate" thinking. The truth is that young children have an amazing capacity to learn complex subject areas, but if early learning opportunities are denied, the power to learn gradually decreases, and may even be lost, as the child grows. But this will have no impact on the NCTM. They're really not concerned about young children learning genuine math. They want them to be "happy". They believe that genuine math is too hard, and learning genuine math is too stressful.

Why Learn Math? The NCTM Says:

"Math" (as defined by the NCTM in chapter 3) is Useful in Everyday Life

The Truth:

  1. The NCTM is only concerned about math needed for everyday living, and they really mean "every" day. All they expect is the subset of arithmetic that's handled by calculators.
  2. Most of math has no "everyday" application. See Why Math is Learned
  3. Who will build those bridges in the twenty-first century? Right now it looks like Asians.

Math Helps "Make Sense" of the Real World

The Truth:

  1. Progressives believe all forms of knowledge are "constructed" through social "experience". More plainly, the only "knowledge" they value is that can be acquired through social discourse. Thus, for progressives to find value in "learning math", they must explain math in terms of social interaction. But, since genuine math has nothing to say about the social world, they must then redefine the meaning of "math" to equate math skills with a subset of social skills.
  2. Math can be used to communicate scientific theory about the "real world", but this requires math above the K-12 level.

How to Learn Math?  The NCTM Says:

A "Broad" Curriculum, Not Carefully Selected Core Math Topics

The Truth:

  1. A very small percent of currently-known math knowledge can be taught during the K-12 years. Kids still need to build a remembered knowledge base starting with math traditionally taught under the headings of arithmetic, algebra, and geometry. This is the core math currently being taught in European and Asian schools.
  2. "Broad goals" mean no specific math content is required. This is part of the NCTM's strategy to avoid accountability. If kids are to "explore broadly", with no two necessarily having the same learning "experience", then standardized tests are ruled out. Progressives hate standardized tests and any other device that will lead to their own accountability.

Discovery Learning: Knowledge Emerges from Experience With Problems, not Through Transmission from Teachers or Books

The Truth:

  1. Progressive educators believe the curriculum should come from the child. They say each individual must discover knowledge for themselves. Not just some knowledge, but all knowledge. They see discovery learning as the exclusive way to learn. "Learning by doing" has been fundamental progressive gospel for eighty years. Now they call it "discovery learning" and claim it's all been proven by "research". Applying this progressive article of faith to math, the NCTM says math knowledge can only result as a byproduct of the effort to solve problems.
  2. Progressives are heavily influenced by "deconstructionism". This literary theory says there's no inherent meaning in literature. Each reader must discover his/her own meaning. This philosophy rejects the traditional belief that different people can come to the same shared understanding of subject matter. They would have us believe that each of us has to "construct our own meaning" for 2 + 2 = 4.
  3. Traditional thinking says you first store math knowledge in your brain, then you apply it to solve problems. The NCTM says this should be "reversed". They say first attempt to solve problems and math knowledge will emerge. Emerge from where? Since they reject transmission from teachers and learning from books, they must believe that math knowledge is somehow already known intuitively and hidden in the brain.
  4. Good math teachers do help kids "discover" math, but not in the vague open-ended way described by the NCTM. The correct approach is to ask questions and present problems that have been carefully chosen to lead students to predetermined learning goals, where such goals are linked to the focused math topics for the grade or course.

Situated learning: The problems for discovery learning should be real-world problems, not textbook problems

The Truth:

  1. For progressive educators, knowledge is only worthwhile if it gives "meaning" to "the real world" and/or is needed for "everyday life". Here they are telling us that knowledge must be "discovered" in real world contexts. Perhaps they don't fully realize that they are rejecting classroom learning.
  2. Math doesn't exist in the real world. It only exists in written form or the minds of humans.
  3. "Our Discovery" above is the only example of a "discovery" offered in the 258 pages of the NCTM Standards. Note that the "proof" is "inductive" via calculator verification. It's apparently too difficult to give a mathematically correct explanation for (A + X) + (B - X) = A + B.

Whole Math, not Step-by-Step Buildup, from Simple to Complex

The Truth:

  1. Here is the attempt to transfer "whole language" to math. Just as progressives deny the structure of the English language and the learning curve for reading, the NCTM wants to deny math structure and the learning curve for math. "Whole" is obviously more absurd in the context of math. Some may be fooled by starting kids with "complex real-world reading", but not by differential equations in the first grade.
  2. A current related insanity is to say math is an "ill-structured domain". But math is the epitome of a structured domain. Learning math involves building a remembered knowledge base, from simple to complex, one math topic at a time, where topics are presented in a coherent step-by-step fashion.

Less is More, not Worrying About Gaps in Knowledge

The Truth:

  1. The "project method", has been progressive gospel since it was first described by Kilpatrick in 1918. It's an extremely inefficient learning method, the ultimate "bits-and-pieces" approach, guaranteed to leave huge gaps in learning. You just can't get to "higher floors" if too many "math knowledge" steps are missing. Too much "less" can quickly equate to zero.
  2. Kids appear to learn through "projects" if they are able to "discover" their parents knowledge. It's great for teachers. Parents do the teaching. The worst part for parents is the admonishment "not to influence your kids with your traditional math". Parents are encouraged to keep quiet and have faith in the "natural" magic of "discovery learning". Smart parents don't listen.



Conceptual Understanding, Not Memorizing and Practicing

The Truth:

  1. You can't understand what you don't remember. Knowledge must be stored in the brain.. Practicing is one way to secure knowledge in memory. Once there, understanding can begin to develop. Each higher level of understanding typically results from new information stored in the brain. "Understanding" is a complex, poorly understood process that involves linking multiple stored "chunks" of knowledge. We have no idea how this magical process occurs. Sometimes we have the experience of the "light bulb turning on", but more frequently understanding gradually improves as new knowledge is added to the constantly growing remembered knowledge base in the brain.
  2. When the NCTM says "decreased attention" they really mean "no attention". They would throw away our primary tool, the unbelievable power of human memory.
  3. Progressive educationists decry "memorizing without understanding". They argue for understanding without remembering. Absurd!

Student-Centered Group Learning: Rely on personal interests and peer authority, not the concerns or authority of teachers, parents, or other adults

The Truth:

  1. "Discovery learning" and "peer authority"? Why are we puzzled by today's kids increasingly discovering drugs, only listening to their peers, and joining gangs?
  2. What makes peers a rich source of math knowledge? The underlying progressive gospel says changing circumstances require each generation to discover its own knowledge and values. They endorse cultural relativism and reject the idea of enduring truths. Remember their leading claim that "math has changed". Unbelievable as it is, it all fits.

Teachers are to Create Exciting Learning Environments for Discovery Learning, not Teach (not Impart Knowledge or Skills)

The Truth:

  1. The NCTM is saying "student teach thyself", or learn from your peers, but please don't ask teachers, your parents, or other adults. Control is "turned over" to students".
  2. Unfortunately, the typical American K-12 math teacher doesn't know much math. This is especially true for K-8 teachers. They really can't be blamed for this. They have been "educated" by a system that doesn't believe in content or knowledge transmission. "Turning over control" to the kids is an understandable cop-out.

Testing to Find Success, not to Detect What Students Have Failed to Learn and not to Improve Teaching Methods and Instructional Materials

The Truth:

  1. Unbelievable as this sounds, it's consistent with the "broad content" and "discovery learning" philosophies of the NCTM. They have no idea what kids might discover! All they can possibly do is somehow describe the "discoveries" and then necessarily bless them as "good".
  2. Students are necessarily "denied access" to a math topic if they don't understand earlier prerequisite math topics. The NCTM can't change this obvious fact.
  3. The first purpose of testing is to detect errors in understanding. The second purpose is to identify defects in teaching materials and/or teaching methods.
  4. Ideally, testing is an integral part of the learning process. Good teachers continually ask questions to test understanding.
Next?

Copyright 1997-2005 William G. Quirk, Ph.D.