The NCTM Calls it "Math"
Chapter 3 of Understanding
the Original NCTM Standards by Bill Quirk
The NCTM Says Traditional
K-12 Math
is Obsolete
The "Introduction" to the NCTM Standards claims that the 54 "standards"
details what mathematics students need to know". Traditionalists will
be
puzzled by this claim. They won't find "focused, specific, basic,
teachable,
and measurable" math content for the K-12 years, and they won't find
traditional
K-12 math. They'll find the NCTM has redefined the meaning of "math",
frequently
emphasizing that traditional K-12 math content should receive
"decreased
attention". Hard as it is to believe, they want to eliminate
traditional
K-12 math content and substitute calculator skills, math appreciation,
and a whole range of general, content-independent skills. They want to
emphasize social goals and psychological considerations, not
traditional
math content.
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:
- 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.
- Math is a constantly growing field, but the foundations of math
haven't
changed.
- 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.
If you buy "traditional K-12 math is obsolete", the NCTM has you set up
to accept their strategy for replacing traditional K-12 math content:
- 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
The NCTM Standards have been developed by progressive "math educators",
not by people with genuine knowledge of mathematics. For eighty years
progressive
educationists have rejected the idea of remembering any domain-specific
knowledge. They say knowledge is changing too fast, and the facts
of today will be obsolete tomorrow. Calculators and computers are the
latest
"proof" of this claim. The NCTM wields them as a double-edged sword,
justifying
the trashing of traditional math and offering the benefit of exciting
"tools"
for bypassing the difficulties of traditional "paper-and-pencil" math.
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:
- Real-world knowledge experts can't afford the time for constantly
looking
up information.
- Expert knowledge is remembered from thousands of sources and
experiences.
It isn't easily available in convenient reference materials.
- "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.
- What fools would belittle and deny the unbelievable power of
human
memory?
In his recent book, The Schools We Need & Why We Don't Have Them,
Professor E. D. Hirsch, Jr. explains the historical process that led to
the current unbelievable mindset of today's K-12 education
establishment:
- Historically, American teacher-training institutions were
primarily
concerned
with teacher mastery of subject matter. Content was primary and
teaching
methods were secondary.
- Teacher-training institutions lost responsibility for content as
they
were
absorbed into universities. For example, the math department
would
now teach math..
- The educationists were left with methods, so they made methods
primary
and began to disparage the importance of remembered content.
- 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.
- 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 result? Today's graduates of American schools of education have
minimal
knowledge of content and they deny the power of human memory. This is
especially
true for K-8 teachers. More importantly, they have been indoctrinated
to
believe that remembering content isn't important.
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:
- 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.
- 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.
- The NCTM never explains why appreciating math is more important
than
learning
math.
- 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:
- 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.
- 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.
- 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:
- 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..
- "Doing math" is not synonymous with these general
content-independent
skills.
- 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:
- 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.
- 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:
- 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?
- 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:
- Progressive educators believe that the only important knowledge
is
already
known intuitively, somehow hidden in the brain. See Discovery
Learning in Chapter 4.
- 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.
- There isn't time to "figure it out each time". What's the point?
Why
not
use the immense power of human memory?
- 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
Note: "Informal" reasoning refers to the use of concrete
materials
(manipulatives).
"Inductive" reasoning refers to generalizing from multiple observations.
- 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:
- 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!
- 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.
- 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.
- 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:
- 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.
- The emphasis on estimation is another time-consuming distraction.
- 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:
- Pattern recognition, like inductive reasoning, is more properly
associated
with science, not math. Here it is one more distraction away from
genuine
math.
- 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:
- 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.
- 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".
Next?
Finding the Context for a
Quote from
the NCTM Standards
Update: The NCTM no longer supports internet
access
to the (original) NCTM Standards. You now need the
hardcopy.
You can no longer search and "Find" as indicated below.
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.
Copyright
1997-2011
William G. Quirk, Ph.D.