The NCTM Calls Them "Standards"

• The Political Side of K-12 Math Standards
• Characteristics of Genuine K-12 Math Standards
1. Characteristics of Individual Math Standards
2. Characteristics of the Complete Set of Math Standards
• The NCTM Standards are not Genuine Math Standards
• Benefits of Genuine K-12 Math Standards
• Added Benefits of National K-12 Math Standards

The Political Side of K-12 Math Standards

1. Standards are about content. K-12 math standards should clearly describe the "K-12 math subset", the specific math content that should be taught grade-by-grade.
2. The NCTM doesn't want genuine content standards. They wants to pass off progressive teaching philosophy as "standards". Today's educationists will work to undermine any strategy that sets the stage for their own accountability. They'll say they support standands, but will continually redefine the meaning of words and drag their feet all the way. They excel at tactics that divide, confuse and delay.
3. You'll hear a lot about calculators! Here's the key point. Calculator skills shouldn't be substituted for mastery of the traditional skills of arithmetic. The NCTM says they agree with this, but they really don't. They push the use of calculators beginning in kindergarten, and nowhere in the 258 pages of the NCTM Standards do they suggest that kids should remember any specific math facts.
4. We can't start with "world class" for every grade. We could start with "world class" first grade standards. For example, we could adopt Japan's first grade math standards as the first version of our national first grade math standards beginning in the fall of 1997. But we can't adopt their fifth grade standards at the same time. If we did, our fifth graders would be lost. They wouldn't have the prerequisite background knowledge built up in the first four grades in Japan.
5. We can evolve "world class" standards. To migrate to "world class" standards we need to annually revise the standards to be increasingly more challenging in every grade. For example, a topic that is initially taught in the eighth grade might eventually migrate down to the fourth grade, say after three annual revisions of the national math standards.

Characteristics of Genuine K-12 Math Standards

1. Characteristics of Individual Math Standards:

• Focused: Each standard covers exactly one math topic, where a math topic is a small closely related set of math facts and math skills
• A math topic is a conceptual "chapter" of math knowledge, not a "book".
• The time to learn a math topic is measured in days or weeks, not years.
• Example: Memorize a set of multiplication facts.
• Example: Add fractions
• Example: Fnd the equation of a straight line when given two points on the line.
• Specific: Each standard should be stated in the most explicit possible way.
• Different K-12 math teachers should easily arrive at the same understanding of the standard.
• Example: "The student will memorize the 25 multiplication facts, from 1x1=1 through 5x5=25".
• Example: "The student will find the equation of a straight line when given the x and y coordinates of two points on the line.
• Basic: Each standard deals with a core knowledge math topic.
• Math needed for everyday life.
• Math needed to develop logical and abstract thinking skills.
• Foundational prerequisite math needed to learn more advanced math
• Math needed to acquire knowledge in other subject areas that utilize math.
• Teachable: Is it possible to teach the topic in a step-by-step manner?
• Without a focused math topic, with clearly identified math facts and math skills, the desired ability is probably not teachable.
• Measurable: Student mastery can be easily evaluated by an objective test.
• Linked to Grade: Each standard should link to exactly one K-12 grade or to a specific named course.
• One teacher is responsible for teaching the standard, measuring student mastery, and taking timely corrective action when mastery is not achieved. If a standard is to be achieved over a multiple grade range, then no specific teacher is responsible for any failure to learn. You have lost teacher accountability. More importantly, you have lost the possibility for timely corrective action.
• If a standard is properly focused, the time needed to master the math topic should never exceed a few weeks. If years are required, the standard is too broad.
• A standard may extend a standard for an earlier grade, but it should not be identical to a standard for an earlier grade or simply rephrase a standard for an earlier grade.
• Concise: Standards should be stated using the minimal number of words needed for clarity.
• Not Redundant: Different standards deal with different math topics.
• Genuine Math: Is it really math, or just called math?
• Would a genuine mathematician recognize it as math? (Note: Generally speaking, K-12 "math educators" don't qualify as "mathematicians".)
• What is the focused math topic that is being taught?
• Genuine math starts with the content traditionally associated with the headings "arithmetic", "algebra", and "geometry". But be aware that the meaning of all three of these traditional terms are being redefined by the math "reformers".

2. Characteristics of the Complete Set of Math Standards:

• Brief: The K-12 math standards document should consist of:
• 30-50 pages
• 2-4 pages of standards per grade or course
• 15-30 standards per grade or course
• Selective: A very small percent of known mathematics can reasonable be covered during the K-12 years.
• The standards should collectively identify the essential core mathematics. The teacher can do more and should have time to do more. The standards identify what all students should learn.
• Coherent Structure: Each standard is properly sequenced after all necessary prerequisite standards.
• The logical structure of a set of standards should be compatible with the structured nature of mathematics. New math knowledge is built on previously learned math knowledge.
• Pedagogically Neutral: The standards should describe the required math content only. They should not specify teaching methods . They should not discuss teaching philosophy.

The NCTM Standards are not Genuine Math Standards

• Not focused
• The NCTM recommends a "broad curriculum", not focused math topics.
• The 54 NCTM "standards" are broad topic headings, such as "Mathematics as Communication".
• Not specific
• The NCTM conspicuously avoids being specific about math content. Each of their 54 "standards" is a multiple-page document.
• The NCTM does appear to believe that kids should learn how to count during the K-4 years, but they never actually state this explicitly. Amazing.
• Not basic
• The NCTM doesn't recognize math as a structured knowledge domain with a core foundational subset of basic domain-specific math facts and math skills. They invite open-ended discovery learning, driven by student interests, not a lesson-by-lesson buildup of core math knowledge.
• Not teachable
• The NCTM rejects the dictionary definition of "teach" ("impart knowledge or skills"). But they still want to call them "teachers". Their roles is to "guide" and create rich enabling environments to excite student interests for discovery learning, with no two students necessarily discovering the same thing..
• Even if the NCTM wanted teachers to teach, their version of K-12 math content is often too broad and ill-defined to be teachable.
• Not measurable
• Because of their fundamental belief in "broad content" and "discovery learning", traditional objective testing must be rejected by the NCTM. All they can do is to attempt to "discover" what each kid has discovered. The NCTM recommends "testing to find success".
• Not linked to grade
• The NCTM standards are not specific about what should happen in each grade. They just discuss general learning goals for grade levels K-4, 5-8, and 9-12.
• Not concise
• As far as writing is concerned, the NCTM rejects "less is more". They constantly repeat words and phrases, often hundreds of times. Examples includes: "vision", "problem solving", "real world", "calculator", "computer", "explore", "experience", "power", "construct", "concrete", "estimate", "measure", and "pattern".
• Redundant
• The excessive redundancy of the NCTM Standards allows the key ideas to captured in the extracted quotes found in Chapter 3 and Chapter 4.
• Not Genuine Math
• This is the topic of Chapter 3.
• Not brief
• The NCTM has produced 59 documents, totaling 258 pages. The NCTM hopes you will be convinced by the weight of the pages.
• Not selective
• The NCTM is repelled by the very thought of a narrow selection of content. They preach broad exploration, not carefully selected math topics.
• Not pedagogically neutral
• The NCTM Standards are not about math, and they are not about standards. They are a vehicle for preaching "progressive" teaching methods.

Benefits of Genuine K-12 Math Standards

1. Clearly identify the subset of essential math knowledge that the student must master.
2. Clearly identify the subset of math knowledge that the teacher must master.
• Clarifies the background knowledge, beyond the student's subset, that the teacher must know in order to teach effectively. The teacher must understand what the student already knows from prior learning, and the teacher must also know how later math knowledge is built on the math knowledge that the teacher is responsible for teaching.
3. Lead to a "multiplier" effect in learning.
• Genuine standards make the learning process much more efficient When a student enters a grade, the teacher knows what the student already knows. Repetition is minimized.
• As remembered math knowledge expands in the brain, new math knowledge is acquired more easily and more rapidly. That is the nature of learning in any knowledge domain. It's slow getting started, but you learn more and more rapidly as the domain-specific knowledge builds in the brain.
• With genuine standards, the bar can be raised higher and higher, without leaving any kid behind.
4. Sets the stage for student accountability.
• Provides the essential prerequisite for objective evaluation of student performance.
• Makes it easy to identify what hasn't been correctly learned.
• Points to defects in teaching materials and methods.
5. Sets the stage for teacher accountability
• Provides the essential prerequisite for objective evaluation of teacher performance.
• Makes each math teacher clearly responsible for a specific subset of K-12 math.
• Makes each math teacher a concerned advocate for improving the effectiveness of math teaching in grades that precede the teacher's own subset of responsibility.
6. Clearly shows parents what is expected.
• Parents can more effectively help their children learn math
• Parents can more effectively evaluate what is happening in the classroom.
7. Encourages math literate citizens, not just parents and teachers, to be actively involved with the math education of all our nation's math illiterate citizens.
• It's not just K-12 kids who need to learn K-12 mathematics. Millions of our citizens need to learn the math that they never learned in misguided "progressive" classrooms.
8. Improves the overall consistency of math teaching.
• One and only one interpretation of each standard by different K-12 math teachers in different schools.
9. Supports collaboration among teachers.
• Encourages the sharing and mutual perfecting of teaching methods:
• Proven best examples to illustrate a specific standard.
• Proven best problems to serve as the focus for teaching a specific math topic.
10. Encourages the development of interactive software to support the teaching of mathematics.
• Currently, the "it must be fun" philosophy of the education establishments dominates, and there is no consensus regarding math content. This handcuffs software development entrepreneurs.
• If software developers are provided with the math content "target", and if they are not required to follow a pre-specified teaching method or teaching philosophy, they will creative effective tutorial software.
• The larger the potential audience, the more entrepreneurs will compete to gain the biggest possible share of the K-12 math software market.

Added Benefits of National K-12 Math Standards

"National" simply means that the same math standards are used everywhere in the United States. "National" does not mean "controlled by the federal government". It just means that each state agrees to use the same set of math standards. For example, all states could agree to begin with the Mathematics Content Standards for California Public Schools  as the first version of national math standards.  From this starting point we can evolve world-class math standards as discussed at the end of the first section of this chapter.

Note: Although "national" would be ideal, "broadly-accepted" would suffice. If enough states agree to share the same set of genuine math standards, this would trigger the "gold rush" described below and initiate the tidal wave that would revolutionize American math education.

National K-12 math standards offer the following added benefits:

1. Avoids the redundant effort involved in developing math standards at the state level.
• There is nothing local about the essential core math knowledge that children should acquire during the K-12 years.
• Disagreements may exist regarding the grade level placement of a specific standard.
• When in doubt, we would be wise to place it in the earlier grade. Contrary to the delay minded "developmentally appropriate" philosophy of the education establishment, our kids are ready to be challenged more and earlier than we have previously recognized.
2. Sets the stage for consistent and focused nationwide preparation of math teachers.
• Extensive teacher retraining and self-study, sharply focused by genuine national math standards, offers our best hope for improving the teaching effectiveness of our current K-12 math teachers.
• Knowledge of math is the essential prerequisite for teaching math. Currently, the typical K-12 math teacher doesn't know much math. Little has been required as part of teacher preparation.
3. Encourages consistent nationwide teaching of mathematics.
• Then, the difficult experience of moving to a new school, possibly in a different state, is softened by the familiar base of math knowledge.
• Note: Most American kids change school at least once during the K-12 years. Many move multiple times. The poor move most frequently.
4. Offers the greatest possible incentives for the development of interactive software to support the teaching of mathematics.
• If a substantial number of states adopts the same set of genuine math standards, this will trigger a "gold rush" for software development entrepreneurs. The resulting "maximum competition" will result in very high quality math tutorial software and the rapid development of the "mathnet" dimension of the internet.

• Chapter 3...... The NCTM Calls it "Math"
• Links to Other Chapters ...... Understanding the Original NCTM Standards
• Understanding the NCTM's Math "Standards" and NCTM "Math Reform" (Home)