The NCTM Calls it "Math"

• The NCTM Says Traditional K-12 Math is Obsolete
• The NCTM Says Math Appreciation is the First Goal
• The NCTM Says Traditional K-12 Math Content Should be Replaced by:
• Calculator Skills and Computer Skills
• General Problems Solving Skills
• General Communication Skills
• The NCTM Says Precision and Exactness Should be Replaced by:
• An Attitude Adjustment About "Correct Answers"
• Intuitive Arithmetic
• Informal and Inductive Reasoning Skills
• Estimation Skills
• Pattern recognition Skills
• The NCTM Still Calls it "Algebra" and "Geometry"

The NCTM Says Traditional K-12 Math is Obsolete

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:

• Calculators and computers have "changed the very nature of the problems important to mathematics and the methods mathematicians use to investigate them".
• "quantitative techniques have permeated almost all intellectual disciplines. However, the fundamental mathematical ideas needed in these areas are not necessarily those studied in the traditional algebra-geometry-precalculus-calculus sequence."
• "For many non mathematicians, arithmetic operations, algebraic manipulations, and geometric terms and theorems constitute the elements of the discipline to be taught in grades K-12. This may reflect the mathematics they studied in school or college rather than a clear insight into the discipline itself."

The Truth:

1. Most of math isn't impacted by calculators and computers.
• But interactive tutorial software and the future mathnet dimension of the internet will completely transform how math is learned.
• Note: The NCTM says "calculators and computers", but they are really only talking about calculators. They see computers as powerful calculators.
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.

• Make math appreciation the primary goal, not building a remembered knowledge base of specific math facts linked to specific math skills.
• Emphasize what can be done with calculators and computers
• Discard topics that don't easily fit.
• Emphasize social goals and psychological considerations.
• Substitute general content-independent "skills" for genuine math knowledge.
• Still call it math and still use traditional math names, but with entirely different meanings for "arithmetic", "algebra", and "geometry".

The Anti-Content, Anti-Memory Progressive Mindset

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?

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.
• In his 1918 article, "Project Method", Kilpatrick argued that knowledge is changing so fast that no specific subject matter should be required in the curriculum.
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 NCTM Says Math Appreciation is the First Goal

• The first goal for all students is "they learn to value mathematics". ( Intro) ** See Quotes
• "Students should have numerous and varied experiences related to the cultural, historical, and scientific evolution of mathematics so that they can appreciate the role of mathematics in the development of our contemporary society." (Intro)

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

• "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." (Intro)
• "Calculators must be accepted at the K-4 level as valuable tools for learning mathematics." (K-4.O)
• "Calculators enable children to compute to solve problems beyond their paper-and-pencil skills." (K-4.8)
• "The calculator renders obsolete much of the complex paper-and-pencil proficiency traditionally emphasized in mathematics courses." (5-8.O)
• "By assigning computational algorithms to calculator or computer processing, this curriculum seeks not only to move students forward but to capture their interest." (9-12.O)

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:
• The student instantly knows that 5 plus 8 equals 13.
• The student instantly knows that 7 times 9 equals 63.
• The student quickly adds 25.3, 144.01, and .004.*
• The student quickly subtracts 2/3 from 5/4.
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

• "Problem solving should be the central focus of the mathematics curriculum" (K-4.1)
• "Mathematical problem solving, in its broadest sense, is nearly synonymous with doing mathematics." (9-12.1)
• "A vital component of problem-solving instruction is having children formulate problems themselves." (K-4.1)
• The problem-solving strategies identified by the NCTM for the K-4 level are "using manipulative materials, using trial and error, making an organized list or table, drawing a diagram, looking for a pattern, and acting out a problem." At the 5-8 level the NCTM adds "guess and check". (K-4.1) and (5-8.1)

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.
• Children can't pose the problems. They don't know what they need to learn.

General Communication Skills, Not the Language and Symbols of Math

• "The communication standard (Standard 2) calls for the integration of language arts as children write and discuss their experiences in mathematics." (K-4.4)
• "Students should be encouraged to explain their reasoning in their own words." (5-8.3)
• "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." (9-12.14)

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 NCTM recommends "decreased attention" for "finding exact forms of answers". (5.8.O)
• "Although written tests structured around a single correct answer can be reliable measures of performance, they offer little evidence of the kinds of thinking and understanding advocated in the Curriculum Standards." (EVAL.2)
• "Students might like mathematics but not display the kinds of attitudes and thoughts identified by this standard. For example, students might like mathematics yet believe that problem solving is always finding one correct answer using the right way. These beliefs, in turn, influence their actions when they are faced with solving a problem. Although such students have a positive attitude toward mathematics, they are not exhibiting essential aspects of what we have termed mathematical disposition." (EVAL.2)

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 NCTM recommends "emphasizing exploratory experiences with numbers that capitalize on the natural insights of children". (K-4.6)
• The NCTM wants children to exercise their "number sense" and "operation sense", but not actually learn how to add, subtract, multiply, and divide without the use of a calculator. (K-4.7 and K-4.8)
• Rather than memorizing the basic addition and subtraction facts, the NCTM encourages "figuring it out each time" using "counting on" and "counting back". (Intro and K-4.6)

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

• For students are the 5-8 level, "concrete experiences should continue to provide the means by which they construct knowledge." (5-8.O)
• It is "essential that in grades 5-8, students explore algebraic concepts in an informal way." (5-8.9)
• "In grades 9-12, the mathematics curriculum should include the informal exploration of calculus concept" (9-12.13)
• "Their previous experience both in and out of school has taught them to accept informal and empirical arguments as sufficient. Students should come to understand that although such arguments are useful, they do not constitute a proof. " (9-12.3)
• "this standard proposes that the organization of geometric facts from a deductive perspective should receive less emphasis." (9-12.7)
• The NCTM says "A mathematician or a student who is doing mathematics often makes a conjecture by generalizing from a pattern of observations made in particular cases (inductive reasoning)" (9-12.3).
• "The assessment of students' ability to reason mathematically should provide evidence that they can use inductive reasoning to recognize patterns and form conjectures." (EVAL.7)

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

• "Instruction should emphasize the development of an estimation mind-set. (K-4.5)
• "Continual emphasis on computational estimation helps children develop creative and flexible thought processes and fosters in them a sense of mathematical power." (K-4.5)
• "When a student wants to know about how long it will take to earn enough baby-sitting money to buy a new bicycle, he or she can estimate the answer." (5-8.7)

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.
• For example, rather than learning how to multiply 1/2 times 1/3 to get the exact answer of 1/6, the student uses the calculator to divide 1 by 2 to get .5, divide 1 by 3 to get .3333, and then multiply these two decimal values to get .1666 as the approximate answer. Of course more decimal places can be used to get a better approximation, say .16666666, but this process will always yield an estimate, not the exact value of 1/6.
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

• "From the earliest grades, the curriculum should give students opportunities to focus on regularities in events, shapes, designs, and sets of numbers. Children should begin to see that regularity is the essence of mathematics." (K-4.13)
• "Identifying patterns is a powerful problem-solving strategy. It is also the essence of inductive reasoning." (5-8.3)
• "Activities in grades 5-8 should build on students' K-4 experiences with patterns. They should continue to emphasize concrete situations that allow students to investigate patterns in number sequences, make predictions, and formulate verbal rules to describe patterns. Learning to recognize patterns and regularities in mathematics and make generalizations about them requires practice and experience. Expanding the amount of time that students have to make this transition to more abstract ways of thinking increases their chances of success." (5-8.9)

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.
• Pattern recognition is not directly teachable, but pattern-recognition skills improve as a byproduct of learning math facts. Thus, if pattern-recognition skills is the goal, teach genuine math facts!

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

• For the 9-12 curriculum, the NCTM recommends "increased attention in algebra" for:
• "The use of computer utilities" (Until end below, quotes are from 9-12.O)
• "Graphing utilities" (calculators or computers) for "solving equations and inequalities"
• and "decreased attention in algebra" for:
• "Word problems by type, such as coin, digit, and work"
• "The use of factoring to solve equations and to simplify rational expressions"
• "Operations with rational expressions"
• "Paper-and-pencil graphing of equations by point plotting."
• and "decreased attention in geometry" for:
• "Euclidean geometry as a complete axiomatic system"
• "Two-column proofs"
• "Analytic geometry as a separate course" (End quotes from 9-12.O)
• Also, about geometry, The NCTM says:
• "Although a facility with the language of geometry is important, it should not be the focus of the geometry program but rather should grow naturally from exploration and experience." (K-4.9)
• In grades 5-8, the " geometry should focus on investigating and using geometric ideas and relationships rather than on memorizing definitions and formulas." (5-8.12)
• "Synthetic geometry at the high school level should focus on more than deductive reasoning and proof." (9-12.8)

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".

• Chapter 4.......The NCTM Calls it "Learning Math"
• Links to Other Chapters ...... Understanding the Original NCTM Standards

Finding the Context for a Quote from the NCTM  Standards

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.