"Goals are broad statements of social intent." (Intro) See Quotes
"'The K-12 standards articulate five general goals for all
students:
(1)
that they learn to value mathematics, (2) that they become confident in
their ability to do mathematics, (3) that they become mathematical
problem
solvers, (4) that they learn to communicate mathematically, and (5)
that
they learn to reason mathematically." (intro)
Note: What the NCTM means by (2) is discussed below. What the
NCTM
means
by (1), (3), (4), and (5) is discussed in chapter 3.
The Truth:
None of these goals deals with genuine math. The NCTM is
interested in
sociology, psychology, and general content-independent skills, not
genuine
K-12 math content. Don't be confused by the NCTM's use of the terms
"mathematics"
and
"mathematically". They're referring to their redefinition of "math", as
described in chapter 3.
K-12 math standards should clearly identify specific academic
content
for each K-12 year.
Once genuine standards are in place, all the resources of our
information
age society can be used to teach the knowledge identified by the
standards.
No longer will we be held hostage by the misguided philosophy of our
current
education establishment.
Building Self-Confidence is More
Important
Than Learning Math
"Affective dimensions of learning play a significant role in, and
must
influence, curriculum and instruction." (K-4.O)
'The curriculum must take seriously the goal of instilling in
students
a sense of confidence in their ability to think and communicate
mathematically."
(K-4.O)
The Truth:
This is the second goal, after math
appreciation. The NCTM is saying self-esteem is more important than
learning math. They never says how this goal can be achieved, and they
never suggest that learning math will help to improve self-confidence.
For the NCTM, self-confidence is the goal, not learning genuine math.
This is fundamental progressive gospel, with "self-esteem" now
called
"self-confidence".
Math standards shouldn't deal with social and psychological
considerations.
"Developmentally
Appropriate"
Learning Takes Considerable Time
"It takes careful planning to create a curriculum that
capitalizes on
children's
intuitive insights and language in selecting and teaching mathematical
ideas and skills. It is clear that children's intellectual, social, and
emotional development should guide the kind of mathematical experiences
they should have in light of the overall goals for learning
mathematics.
The notion of a developmentally appropriate curriculum is an important
one." (K-4.O)
"A developmentally appropriate curriculum encourages the
exploration of
a wide variety of mathematical ideas in such a way that children retain
their enjoyment of, and curiosity about, mathematics. It incorporates
real-world
contexts, children's experiences, and children's language in developing
ideas. It recognizes that children need considerable time to construct
sound understandings and develop the ability to reason and communicate
mathematically." (K-4.O)
The Truth:
Why hasn't Johnny learned arithmetic by the eighth grade? The
answer?
The
NCTM has achieved their goal if Johnny appears to be "happy". Learning
math is not required. Kids must be free to develop at their own
"natural
pace". They're fragile and can't be rushed. Current "happiness" is the
goal. There's no concern about what will happen to them after the K-12
years.
Recently, genuine research in cognition has disproved
"developmentally
appropriate" thinking. The truth is that young children have an amazing
capacity to learn complex subject areas, but if early learning
opportunities
are denied, the power to learn gradually decreases, and may even be
lost,
as the child grows. But this will have no impact on the NCTM. They're
really
not concerned about young children learning genuine math. They want
them
to be "happy". They believe that genuine math is too hard, and learning
genuine math is too stressful.
Why Learn Math? The NCTM Says:
"Math" (as defined by the NCTM in chapter 3) is
Useful
in Everyday Life
"We do not assert that informational knowledge has no value, only
that
its value lies in the extent to which it is useful in the course of
some
purposeful activity." (Intro)
"mathematics should not be disconnected from students' daily
lives."
(5-8.4)
the curriculum "must emphasize the usefulness of mathematics, and
must
foster a positive disposition toward mathematics." (5.8.O)
The Truth:
The NCTM is only concerned about math needed for everyday living,
and
they
really mean "every" day. All they expect is the subset of arithmetic
that's
handled by calculators.
Who will build those bridges in the twenty-first century? Right
now it
looks like Asians.
Math Helps "Make Sense" of the Real World
The NCTM says students must "explore and make sense of their
world"
(5-8.O)
The NCTM believes "mathematics evolves naturally from problem
situations
that have meaning to children and are regularly related to their
environment."
(K-4.1)
The Truth:
Progressives believe all forms of knowledge are "constructed"
through
social
"experience". More plainly, the only "knowledge" they value is that can
be acquired through social discourse. Thus, for progressives to find
value
in "learning math", they must explain math in terms of social
interaction.
But, since genuine math has nothing to say about the social world, they
must then redefine the meaning of "math" to equate math skills with a
subset
of social skills.
Math can be used to communicate scientific theory about the "real
world",
but this requires math above the K-12 level.
How to Learn Math? The
NCTM
Says:
A "Broad" Curriculum, Not Carefully
Selected
Core Math Topics
"The K-4 curriculum should include a broad range of content."
(K-4.O)
"Students entering grade 9 will have experienced mathematics in
the
context
of the broad, rich curriculum outlined in the K-8 standards." (9-12.O)
"High school mathematics instruction must adopt broader goals for
all
students."
(9-12.O)
Evaluators should look for increased attention for "focusing on a
broad
range of mathematical tasks and taking a holistic view of mathematics"
(Eval.O)
The Truth:
A very small percent of currently-known math knowledge can be
taught
during
the K-12 years. Kids still need to build a remembered knowledge base
starting
with math traditionally taught under the headings of arithmetic,
algebra,
and geometry. This is the core math currently being taught in European
and Asian schools.
"Broad goals" mean no specific math content is required. This is
part
of
the NCTM's strategy to avoid accountability. If kids are to "explore
broadly",
with no two necessarily having the same learning "experience", then
standardized
tests are ruled out. Progressives hate standardized tests and any other
device that will lead to their own accountability.
Discovery Learning: Knowledge
Emerges
from Experience With Problems, not Through Transmission from Teachers
or
Books
"Traditional teaching emphases on practice in manipulating
expressions
and practicing algorithms as a precursor to solving problems ignore the
fact that knowledge often emerges from the problems. This suggests that
instead of the expectation that skill in computation should precede
word
problems, experience with problems helps develop the ability to
compute.
Thus, present strategies for teaching may need to be reversed;
knowledge
often should emerge from experience with problems." (Intro)
before young children are taught addition and subtraction, they
can
already
solve most addition and subtraction problem. (Intro)
Children need to "construct their own meaning". (Intro)
"knowing" mathematics is "doing" mathematics. A person gathers,
discovers,
or creates knowledge in the course of some activity having a purpose.
This
active process is different from mastering concepts and procedures"
(Intro)
"instruction should persistently emphasize "doing" rather than
"knowing
that"." (Intro)
For the K-4 grades, the NCTM recommends "decreased attention" for
"teaching
by telling." (K-4.O)
The Truth:
Progressive educators believe the curriculum should come from the
child.
They say each individual must discover knowledge for themselves. Not
just
some knowledge, but all knowledge. They see discovery learning
as
the exclusive way to learn. "Learning by doing" has been fundamental
progressive
gospel for eighty years. Now they call it "discovery learning" and
claim
it's all been proven by "research". Applying this progressive article
of
faith to math, the NCTM says math knowledge can only result as a
byproduct
of the effort to solve problems.
Don't be fooled by "knowledge often should emerge" and
"decreased
attention", progressive educators really believe knowledge can only
emerge
from efforts to solve problems and there should be no teaching by
telling.
If they deny such extremism, ask them for examples of knowledge that
can
be acquired in other ways, and ask for examples of knowledge that
teachers
should "transmit" to students. Ask if kids should memorize the
multiplication
table. Expect evasion, deception, or silence.
Progressives are heavily influenced by "deconstructionism". This
literary
theory says there's no inherent meaning in literature. Each reader must
discover his/her own meaning. This philosophy rejects the traditional
belief
that different people can come to the same shared understanding of
subject
matter. They would have us believe that each of us has to "construct
our
own meaning" for 2 + 2 = 4.
Traditional thinking says you first store math knowledge in your
brain,
then you apply it to solve problems. The NCTM says this should be
"reversed".
They say first attempt to solve problems and math knowledge will
emerge.
Emerge from where? Since they reject transmission from teachers and
learning
from books, they must believe that math knowledge is somehow already
known intuitively and hidden in the brain.
This explains why the NCTM emphasizes intuition about numbers,
operations,
and space, rather than helping kids to learn and remember basic facts
of
arithmetic, algebra, and geometry.
How does the NCTM get around the fact that kids don't
"naturally" know
the multiplication table? They declare that "math" isn't about
remembering
such facts. They say "math" is calculator skills, math appreciation,
and
the general content-independent skills of chapter
3.
Genuine math knowledge is not in the brain at birth. Someone
has to
tell
you that 1 + 1 = 2. You can only learn math from teachers and books.
Good math teachers do help kids "discover" math, but not in
the
vague
open-ended way described by the NCTM. The correct approach is to
ask
questions and present problems that have been carefully chosen to lead
students to predetermined learning goals, where such goals are linked
to
the focused math topics for the grade or course.
Situated learning: The problems for
discovery
learning should be real-world problems, not textbook problems
"All mathematics should be studied in contexts that give the
ideas and
concepts meaning. (5-8.O)
"Students should have many experiences in creating problems from
real-world
activities." (K-4.1)
It is "essential that the instructional program provide
opportunities
for
students to generate procedures. Such opportunities should dispel the
belief
that procedures are predetermined sequences of steps handed down by
some
authority (e.g., the teacher or the textbook)." (EVAL.9)
"They were using a calculator to explore number relationships
when they
noticed that if one addend is decreased by any amount and another
addend
is increased by the same amount, their sum remains the same. After
checking
their conjecture with a variety of numbers, they recorded it as a
discovery
so that it could be shared with the rest of the class." (K-4.3)
"Our Discovery: When you add, if you make one part bigger and
the other
part gets the same amount smaller, you always get the same answer."
(K-4.3)
The Truth:
For progressive educators, knowledge is only worthwhile if it
gives
"meaning"
to "the real world" and/or is needed for "everyday life". Here they are
telling us that knowledge must be "discovered" in real world
contexts.
Perhaps they don't fully realize that they are rejecting classroom
learning.
Math doesn't exist in the real world. It only exists in written
form or
the minds of humans.
"Our Discovery" above is the only example of a "discovery"
offered in
the
258 pages of the NCTM Standards. Note that the "proof" is "inductive"
via
calculator verification. It's apparently too difficult to give a
mathematically
correct explanation for (A + X) + (B - X) = A + B.
Whole Math, not Step-by-Step Buildup, from
Simple
to Complex
"mathematics must be approached as a whole. Concepts, procedures,
and
intellectual
processes are interrelated. In a significant sense, "the whole is
greater
than the sum of its parts." " (Intro)
The "broad range of topics" of the 5-8 curriculum "should be
taught as
an integrated whole, not as isolated topics" (5.8.O)
"Students should be given tasks that are challenging and
complex."
(EVAL.3)
The Truth:
Here is the attempt to transfer "whole language" to math. Just as
progressives
deny the structure of the English language and the learning curve for
reading,
the NCTM wants to deny math structure and the learning curve for math.
"Whole" is obviously more absurd in the context of math. Some may be
fooled
by starting kids with "complex real-world reading", but not by
differential
equations in the first grade.
"Complex" here is really just talk. Presumably, "whole
language"
reading
teachers can read complex texts, but few of our K-12 math teachers know
about complex math.
What does the NCTM mean by "complex"? Wallpapering a room is as
complex
as it gets for them. Remember their universe of concern is "everyday"
life.
"
A current related insanity is to say math is an "ill-structured
domain".
But math is the epitome of a structured domain. Learning math involves
building a remembered knowledge base, from simple to complex, one math
topic at a time, where topics are presented in a coherent step-by-step
fashion.
Less is More, not Worrying About Gaps in
Knowledge
"Although quantitative considerations have frequently dominated
discussions
in recent years, qualitative considerations have greater significance.
Thus, how well children come to understand mathematical ideas is far
more
important than how many skills they acquire." (K-4.O)
"Real-world problems often require a substantial investment of
time.
Students
should be encouraged to explore some problems as extended projects that
can be worked on for hours, days, or longer." (5-8.1)
The Truth:
The "project method", has been progressive gospel since it was
first
described
by Kilpatrick in 1918. It's an extremely inefficient learning method,
the
ultimate "bits-and-pieces" approach, guaranteed to leave huge gaps in
learning.
You just can't get to "higher floors" if too many "math knowledge"
steps
are missing. Too much "less" can quickly equate to zero.
Kids appear to learn through "projects" if they are able
to
"discover"
their parents knowledge. It's great for teachers. Parents do the
teaching.
The worst part for parents is the admonishment "not to influence your
kids
with your traditional math". Parents are encouraged to keep quiet and
have
faith in the "natural" magic of "discovery learning". Smart parents
don't
listen.
Conceptual Understanding, Not
Memorizing
and Practicing
"The curriculum should focus on the development of understanding,
not
on
the rote memorization of formulas." (5-8.13)
For the 5-8 curriculum, the NCTM recommends "decreased attention"
for
memorizing
and practicing. (5-8.O)
"The 9-12 standards call for a shift in emphasis from a
curriculum
dominated
by memorization of isolated facts and procedures and by proficiency
with
paper-and-pencil skills to one that emphasizes conceptual
understandings."
(9-12.O)
The Truth:
You can't understand what you don't remember. Knowledge must be
stored
in the brain.. Practicing is one way to secure knowledge in memory.
Once
there, understanding can begin to develop. Each higher level of
understanding
typically results from new information stored in the brain.
"Understanding"
is a complex, poorly understood process that involves linking multiple
stored "chunks" of knowledge. We have no idea how this magical process
occurs. Sometimes we have the experience of the "light bulb turning
on",
but more frequently understanding gradually improves as new knowledge
is
added to the constantly growing remembered knowledge base in the brain.
When the NCTM says "decreased attention" they really mean "no
attention".
They would throw away our primary tool, the unbelievable power of human
memory.
Progressive educationists decry "memorizing without
understanding".
They
argue for understanding without remembering. Absurd!
Student-Centered Group Learning: Rely
on personal
interests and peer authority, not the concerns or authority of
teachers,
parents, or other adults
"This approach instills in students an understanding of the value
of
independent
learning and judgment and discourages them from relying on an outside
authority
to tell them whether they are right or wrong." (5-8.O)
"Groups might first share their questions with the whole class
and
then,
in small groups, decide which questions they wish to investigate. Such
situations allow students to formulate questions based on their own
interests."
(5-8.1)
"Students learn from the " thought processes of their peers" and
"peer
instruction". (9-12.O)
The Truth:
"Discovery learning" and "peer authority"? Why are we puzzled by
today's
kids increasingly discovering drugs, only listening to their peers, and
joining gangs?
What makes peers a rich source of math knowledge? The underlying
progressive
gospel says changing circumstances require each generation to discover
its own knowledge and values. They endorse cultural relativism and
reject
the idea of enduring truths. Remember their leading claim that "math
has changed". Unbelievable as it is, it all fits.
Teachers are to Create Exciting
Learning
Environments for Discovery Learning, not Teach (not Impart Knowledge or
Skills)
"Teachers need to create an environment that encourages children
to
explore."
(K-4.O)
"Technology can foster environments in which students' growing
curiosity
can lead to rich mathematical invention. In these environments, the
control
of exploring mathematical ideas is turned over to students."
(5-8.3)
The Truth:
The NCTM is saying "student teach thyself", or learn from your
peers,
but
please don't ask teachers, your parents, or other adults. Control is
"turned
over" to students".
Unfortunately, the typical American K-12 math teacher doesn't
know much
math. This is especially true for K-8 teachers. They really can't be
blamed
for this. They have been "educated" by a system that doesn't believe in
content or knowledge transmission. "Turning over control" to the kids
is
an understandable cop-out.
Since our teachers don't know much math, but math knowledge
must
somehow
be transmitted to our kids, there is only one general solution on the
horizon.
If we intelligently invest in interactive tutorial software and the
mathnet
dimension of the internet, we can return to leading the world in the
quality
of K-12 math.
Testing to Find Success, not to Detect What
Students
Have Failed to Learn and not to Improve Teaching Methods and
Instructional
Materials
The NCTM recommends "assessing what students know and how they
think
about
mathematics" , not assessing what students do not know. 9-12.O
"The main purpose of evaluation, as described in these standards,
is to
help teachers better understand what students know." (EVAL.O)
"Assessment should examine students' disposition toward
mathematics, in
particular their confidence in doing mathematics and the extent to
which
they value mathematics." (EVAL.4)
"No student should be denied access to the study of one topic
because
he
or she has yet to master another." (5.8.O)
The Truth:
Unbelievable as this sounds, it's consistent with the "broad
content"
and
"discovery learning" philosophies of the NCTM. They have no idea what
kids
might discover! All they can possibly do is somehow describe the
"discoveries"
and then necessarily bless them as "good".
Students are necessarily "denied access" to a math topic if they
don't
understand earlier prerequisite math topics. The NCTM can't change this
obvious fact.
The first purpose of testing is to detect errors in
understanding. The
second purpose is to identify defects in teaching materials and/or
teaching
methods.
Ideally, testing is an integral part of the learning process.
Good
teachers
continually ask questions to test understanding.