The NCTM Calls Them "Standards"
Chapter 2 of Understanding
the Original NCTM Standards By Bill Quirk
The Political
Side of
K-12 Math Standards
- 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.
- 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.
- 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.
- 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.
- 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
The NCTM
Standards claim to describe K-12 math
content.
What kind of a description has the NCTM given? Note: The links
in
this section will take you to sections of chapters 3 and 4.
- 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
- 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
- Clearly identify the subset of essential math knowledge that the
student
must master.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- Improves the overall consistency of math teaching.
- One and only one interpretation of each standard by different
K-12 math
teachers in different schools.
- 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.
- 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:
- 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.
- 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.
- 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.
- 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.
Next?
Copyright
1997-2011
William G. Quirk, Ph.D.