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