The NCTM Calls Them "Standards"
Chapter 2 of Understanding
the Original NCTM Standards By Bill Quirk
The NCTM Standards are
Genuine K-12 Math
of the Complete
Set of Math Standards
K-12 Math Standards
- Standards are about content. K-12 math standards should
describe the "K-12 math subset", the specific math content that should
be taught grade-by-grade.
- The NCTM doesn't want genuine content standards. They
off progressive teaching philosophy as "standards". Today's
will work to undermine any strategy that sets the stage for their own
They'll say they support standands, but will continually redefine the
of words and drag their feet all the way. They excel at tactics that
confuse and delay.
- You'll hear a lot about calculators! Here's the key
skills shouldn't be substituted for mastery of the traditional
of arithmetic. The NCTM says they agree with this, but they really
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.
- We can't start with "world class" for every grade. We
with "world class" first grade standards. For example, we could adopt
first grade math standards as the first version of our national first
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
be lost. They wouldn't have the prerequisite background knowledge built
up in the first four grades in Japan.
- We can evolve "world class" standards. To migrate to
standards we need to annually revise the standards to be increasingly
challenging in every grade. For example, a topic that is initially
in the eighth grade might eventually migrate down to the fourth grade,
say after three annual revisions of the national math standards.
K-12 Math Standards
of Individual Math Standards:
- Focused: Each standard covers exactly one math topic,
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
- The time to learn a math topic is measured in days or weeks,
- Example: Memorize a set of multiplication facts.
- Example: Add fractions
- Example: Fnd the equation of a straight line when given two
- Specific: Each standard should be stated in the most
- Different K-12 math teachers should easily arrive at the same
of the standard.
- Example: "The student will memorize the 25 multiplication
- Example: "The student will find the equation of a straight line
the x and y coordinates of two points on the line.
- Basic: Each standard deals with a core knowledge math
- Math needed for everyday life.
- Math needed to develop logical and abstract thinking skills.
- Foundational prerequisite math needed to learn more advanced
- Math needed to acquire knowledge in other subject areas that
- Teachable: Is it possible to teach the topic in a
- Without a focused math topic, with clearly identified math
skills, the desired ability is probably not teachable.
- Measurable: Student mastery can be easily evaluated by
- Linked to Grade: Each standard should link to exactly
or to a specific named course.
- One teacher is responsible for teaching the standard, measuring
mastery, and taking timely corrective action when mastery is not
If a standard is to be achieved over a multiple grade range, then no
teacher is responsible for any failure to learn. You have lost teacher
accountability. More importantly, you have lost the possibility for
- If a standard is properly focused, the time needed to master
should never exceed a few weeks. If years are required, the standard is
- A standard may extend a standard for an earlier grade, but it
be identical to a standard for an earlier grade or simply rephrase a
for an earlier grade.
- Concise: Standards should be stated using the minimal
words needed for clarity.
- Not Redundant: Different standards deal with different
- Genuine Math: Is it really math, or just called math?
- Would a genuine mathematician recognize it as math? (Note:
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
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".
Characteristics of the
Complete Set of Math Standards:
- Brief: The K-12 math standards document should consist
- 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
be covered during the K-12 years.
- The standards should collectively identify the essential core
The teacher can do more and should have time to do more. The
identify what all students should learn.
- Coherent Structure: Each standard is properly sequenced
necessary prerequisite standards.
- The logical structure of a set of standards should be
structured nature of mathematics. New math knowledge is built on
learned math knowledge.
- Pedagogically Neutral: The standards should describe the
math content only. They should not specify teaching methods .
should not discuss teaching philosophy.
The NCTM Standards are not Genuine
Standards claim to describe K-12 math
What kind of a description has the NCTM given? Note: The links
this section will take you to sections of chapters 3 and 4.
- Not focused
- The NCTM recommends a "broad
not focused math topics.
- The 54 NCTM "standards" are broad topic headings, such as
- Not specific
- The NCTM conspicuously avoids being specific about math
their 54 "standards" is a multiple-page document.
- The NCTM does appear to believe that kids should learn how to
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
They invite open-ended discovery
learning, driven by student
not a lesson-by-lesson buildup of core math knowledge.
- Not teachable
- The NCTM rejects the dictionary definition of "teach" ("impart
or skills"). But they still want to call them "teachers". Their roles
to "guide" and create
environments to excite student interests for discovery learning,
no two students necessarily discovering the same thing..
- Even if the NCTM wanted teachers to teach, their
of K-12 math content is often too broad and ill-defined to be
- Not measurable
- Because of their fundamental belief in "broad content" and
traditional objective testing must be rejected by the NCTM. All they
do is to attempt to "discover" what each kid has discovered. The NCTM
to find success".
- Not linked to grade
- The NCTM standards are not specific about what should happen in
They just discuss general learning goals for grade levels K-4, 5-8, and
- Not concise
- As far as writing is concerned, the NCTM rejects "less is
repeat words and phrases, often hundreds of times. Examples includes:
"problem solving", "real world", "calculator", "computer", "explore",
"power", "construct", "concrete", "estimate", "measure", and "pattern".
- The excessive redundancy of the NCTM Standards allows the key
captured in the extracted quotes found in Chapter
3 and Chapter 4.
- Not Genuine Math
- Not brief
- The NCTM has produced 59 documents, totaling 258 pages. The
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
They preach broad exploration, not carefully selected math topics.
- Not pedagogically neutral
- The NCTM Standards are not about math,
not about standards. They are a vehicle for preaching "progressive"
K-12 Math Standards
- Clearly identify the subset of essential math knowledge that the
- Clearly identify the subset of math knowledge that the teacher
Lead to a "multiplier" effect in learning.
- Clarifies the background knowledge, beyond the student's
teacher must know in order to teach effectively. The teacher must
what the student already knows from prior learning, and the teacher
also know how later math knowledge is built on the math knowledge that
the teacher is responsible for teaching.
Sets the stage for student accountability.
- Genuine standards make the learning process much more efficient
student enters a grade, the teacher knows what the student already
Repetition is minimized.
- As remembered math knowledge expands in the brain, new
math knowledge is acquired more easily and more rapidly. That is
nature of learning in any knowledge domain. It's slow getting started,
but you learn more and more rapidly as the domain-specific knowledge
in the brain.
- With genuine standards, the bar can be raised higher and
leaving any kid behind.
Sets the stage for teacher accountability
- Provides the essential prerequisite for objective evaluation of
- Makes it easy to identify what hasn't been correctly learned.
- Points to defects in teaching materials and methods.
- Provides the essential prerequisite for objective evaluation of
- Makes each math teacher clearly responsible for a specific
Clearly shows parents what is expected.
- Makes each math teacher a concerned advocate for improving the
of math teaching in grades that precede the teacher's own subset of
Encourages math literate citizens, not just parents and teachers,
actively involved with the math education of all our nation's math
- Parents can more effectively help their children learn math
- Parents can more effectively evaluate what is happening in the
Improves the overall consistency of math teaching.
- It's not just K-12 kids who need to learn K-12 mathematics.
our citizens need to learn the math that they never learned in
Supports collaboration among teachers.
- One and only one interpretation of each standard by different
teachers in different schools.
Encourages the development of interactive software to
- Encourages the sharing and mutual perfecting of teaching
- Proven best examples to illustrate a specific standard.
- Proven best problems to serve as the focus for teaching a
- Currently, the "it must be fun" philosophy of the education
dominates, and there is no consensus regarding math content. This
software development entrepreneurs.
- If software developers are provided with the math content
if they are not required to follow a pre-specified teaching method or
philosophy, they will creative effective tutorial software.
- The larger the potential audience, the more entrepreneurs will
to gain the biggest possible share of the K-12 math software market.
of National K-12 Math Standards
"National" simply means that the same math standards are used
in the United States. "National" does not mean "controlled by the
government". It just means that each state agrees to use the same set
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
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
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:
- Avoids the redundant effort involved in developing math standards
Sets the stage for consistent and focused nationwide preparation
- There is nothing local about the essential core math knowledge
should acquire during the K-12 years.
- Disagreements may exist regarding the grade level placement of
- When in doubt, we would be wise to place it in the earlier
to the delay minded "developmentally appropriate" philosophy of the
establishment, our kids are ready to be challenged more and earlier
we have previously recognized.
Encourages consistent nationwide teaching of mathematics.
- Extensive teacher retraining and self-study, sharply focused
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
the typical K-12 math teacher doesn't know much math. Little has been
as part of teacher preparation.
Offers the greatest possible incentives for the development of
software to support the 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
Many move multiple times. The poor move most frequently.
- If a substantial number of states adopts the same set of
this will trigger a "gold rush" for software development entrepreneurs.
The resulting "maximum competition" will result in very high quality
tutorial software and the rapid development of the "mathnet" dimension
of the internet.
William G. Quirk, Ph.D.