The Truth About The New Jersey
An Analysis of the New Jersey Math Standards (NJMS)
by Bill Quirk (E-Mail:
The New Jersey Math Standards Are
Based On the NCTM Standards
Jersey Math Standards (NJMS) are a "child" of The NCTM
Standards, from The National Council of Teachers of Mathematics
Similar to the parent, they fail to identify the specific,
math content that should be taught and learned during the K-12 years.
they promote the NCTM's "progressive" program for replacing traditional
K-12 math with "manipulatives", calculators, "guess and check",
and content-independent "process skills". These NJMS substitution
are disguised as 16 so-called "content standards", each with multiple
process indicators" (CPIs) listed for the ends of grades 4, 8, and 12.
Yes, New Jersey high school graduates will be able to guess the
to 3x + 2 = 17, and they will be able to use a calculator to estimate
solution to 6x +2 = 22, but don't be surprised if they can't find the
answer in the second case. Expect to hear that "exact answers" aren't
and that modern math is all about calculators. Don't be surprised when
they reach for a calculator to find 10% of 300, or to find the new
price of an $8 item, now offered at a 25% discount.
The NJMS achieves its lean 26-page appearance by speaking in
and by suppressing the underlying "progressive" philosophy found in the
Standards. That's all found in the 670-page New
Jersey Mathematics Curriculum Framework (NJMCF), a 1500
cookbook for math self-discovery. Following the lead of the NCTM, the
"progressive" axioms have been assumed by the writers of the NJMCF:
- Belief that children must be allowed to follow their own
personally discover the math knowledge that they find interesting and
to their own lives.
Belief that children must "construct" mathematical knowledge for
- Rejection of the concept of a common core of basic math
all children should learn during the K-12 years
Belief that all knowledge must be acquired as a byproduct of
in real-world settings.
- Rejection of teacher-directed knowledge transmission.
Belief in the primary importance of general, content-independent
- Rejection of classroom learning.
Belief that calculators have fundamentally changed the nature of
- Rejection of the need to remember any specific math content.
Belief that learning must always be an enjoyable, happy
knowledge emerging naturally from games and group activities.
- Rejection of the need to acquire traditional paper-and-pencil
When it comes to the learning math, progressives preach their gospel of
"discovering math through problem solving". You may think this
to the traditional process whereby teachers ask questions and present
which have been carefully chosen to lead students to discover
math knowledge. Not so! Progressives preach open-ended "exploration",
no expectation that different kids will "discover" the same thing.
about a careful step-by-step buildup of core math knowledge that all
learn to understand in the same correct way. Progressive educationists
believe that each child must "construct their own meaning", with their
own personal version of mathematical knowledge somehow emerging from
to solve complex, real-world problems, with the further complexity that
the problems must be chosen by the students, based on their personal
Progressives don't believe it's right to pre-specify what kids should
and they don't believe that all kids should be required to learn the
content. This in turn forces them to redefine the meaning of
to equate it with "finding out what each kid has discovered", rather
identifying what each student has failed to learn. More generally, New
Jersey math educationists have redefined the meaning of
"math", and "learning math".
- Rejection of any attempt to challenge a child to work harder.
New Jersey math educators want you to believe that their philosophy
is backed up by recent research in cognitive psychology. Don't buy it.
They've been trying to sell this philosophy for 80 years. If they talk
excitedly, using high-sounding terms like "constructivism", "situated
"holistic learning", "developmentally appropriate", and
you'll know you're experiencing a strange encounter with a member of
progressive educationist cult. Good luck!
The good news is that genuine researchers are also appalled by the
of today's progressive educationists. This truth was recently explained
by E. D. Hirsch, Jr. in his April 10, 1997 Address
to The California State Board of Education. In his presentation
Hirsch discussed the difficulty of publishing an important research
and Misapplications of Cognitive Psychology to Mathematics Education,
even though one of the co-authors, Herbert A. Simon, is a winner of the
Nobel Prize, and the other co-authors, John R. Anderson and Lynne M.
have the highest respect of those who do genuine research in cognitive
psychology. Most progressive educationists don't know about these
and their research. Those who do know don't want you to know.
Rather than wading through the 670-page NJMCF,
you can find what you need to know by reading this document, together
the linked sections from The Truth About the
Standards. This will be more than enough for you to fully
how New Jersey educationists want to emphasize "progressive" social
and the fundamental progressive gospel of "discovering math through
solving". Note: At any time you can use the NJMCF
links to check the context of NJMCF
quotes. Page references are given.
The NJMS Are Not Genuine Math
fail to qualify as genuine math standards when evaluated according to
of Genuine K-12 Math Standards listed in Chapter
2 of The Truth About The NCTM Standards.
Genuine math standards should clearly describe the grade-by-grade math
content for each K-12 year. The NJMSand
fail to do this. They are:
- Not focused
- The 16 NJMS "content standards" are very general learning
focused math topics. Each NJMS "standard" has an associated list of
Progress Indicators" (CPIs) which still fail to identify focused math
But this is the lowest level of detail found in the NJMS! Consider
very general nature of the following CPIs, listed for the end of
(Note: The NJMS "content standards" are numbered from 4.1
to 4.16 in the NJMS, and 1 to 16 in the NJMCF. We will use the NJMCF 1
to 16 approach. Also, the notation N.K indicates CPI K for
- CPI 1.1: Use discovery-oriented, inquiry-based, and
to investigate and understand mathematical content appropriate to early
CPI 2.2: Identify and explain key mathematical concepts, and
using oral, written, concrete, pictorial, and graphical methods.
CPI 3.1: View mathematics as an integrated whole rather than
of disconnected topics and rules.
CPI 4.1: Make educated guesses and test them for correctness.
CPI 5.2: Select and use calculators, software, manipulatives,
tools based on their utility and limitations and on the problem
CPI 6.1 Use real-life experiences, physical materials, and
construct meaning for whole numbers, commonly used fractions, and
CPI 7.5 Investigate and predict the results of combining,
and changing shapes.
CPI 8.5: Use a variety of mental computation and
CPI 10.4: Explore, construct, and use a variety of estimation
CPI 11.5: Observe and recognize examples of patterns,
functions in other disciplines and contexts.
CPI 12.7: Make predictions based on intuitive, experimental,
The NJMCF goes one level below the CPI, associating multiple
and games with each CPI. But there's still an overall failure to
focused math topics, and there's frequently no evident reason as to why
a particular activity or game is associated with a particular CPI.
the NJMCF "activities" are supposed to help New Jersey kids discover
math. What math? That's usually not clear.
- CPI 1.10 substitutes "content appropriate to the middle
1.15 substitutes "content appropriate to the high school grades".
- The Truth: The detailed identification of
appropriate" for each grade is exactly the role of math standards!
- New Jersey educationists conspicuously avoid being specific
content for the K-12 years. They believe that each kid should be
to explore broadly.
- New Jersey math educators don't believe in identifying and
math knowledge. Instead, they endorse broad, discovery
learning (Link to Chapter 4 of The Truth About The NCTM Standards)
- The NJMS and NJMCF are too broad and unfocused to be teachable.
isn't a term that has meaning for progressives. Their endorsement of
learning implies a corresponding rejection of teacher-directed
transmission. "Progressive" educators believe it is wrong for teachers,
or anyone else, to decide what kids should learn. They say the main
of the teacher is to create
rich enabling environments (Link to Chapter 4 of The Truth About
Not linked to grade
- Because of their fundamental belief in "broad content" and
traditional objective testing is ruled out. All "teachers" can do is to
attempt to "discover" what each kid has personally discovered. That is,
all they can do is "test to find
to Chapter 4 of The Truth About The NCTM Standards)
Concise, to a fault
- The NJMS and NJMCF are not specific about what should happen in
They discuss general learning goals for the ends of grade 4, 8, and 12.
There is no possibility for either student or teacher accountability.
- CPI's are concisely stated. That is, they do "express much in
The problem is that they express far too much, using general words.
- Consider the redundancy in the following examples:
Not genuine math
- CPI 8.5: Use a variety of mental computation and estimation
- CPI 8.6: Select and use appropriate computational methods
estimation, paper-and-pencil, and calculator methods and check for
- CPI 8.10: Develop, apply, and explain procedures for
with whole numbers, fractions, decimals, integers, and rational numbers.
- CPI 10.4: Explore, construct, and use a variety of estimation
- CPI 10.6: Determine the reasonableness of an answer by
- CPI 10.7: Apply estimation in working with quantities,
computation, and problem solving.
- CPI 10.8: Develop, apply, and explain a variety of different
strategies involving quantities and measurement.
- This is the most alarming problem and the topic of the
- Relative to the 217 page NCTM parent, the 26 page NJMS is
fair comparison is to include the 670-page NJMCF, for a total 696 pages.
Not pedagogically neutral
- Today's progressive educationists are repelled by the very
thought of a
narrow selection of content. They preach broad, open-ended exploration,
not carefully selected math topics.
- The NJMS don't explicitly preach "progressive" teaching
assume them! All the key words are there. The detailed progressive
and "teaching" methods are found in the 670-page NJMCF
The NJMS Offer A Program For
Traditional K-12 Math With Calculators, Math Appreciation, and
Chapter 3 and chapter 4
of The Truth About The NCTM Standards
how the NCTM wants to replace traditional core math knowledge with
math appreciation, and general, content-independent "process skills".
"progressive" objectives are also the central message of The NJMSand
NJMCF. This becomes evident when we see the NJMCF "details"
for the 16 New Jersey Math Content "Standards".
Note: Links below are to related sections of The
Truth About The NCTM Standards.
- Problem Solving (The
standard" and the first "process skill")
"content standard" and the second "process skill")
- The NJMCF say that the problem-solving "process skills" to be
"making a list, making a chart or a table, drawing a diagram, making a
model, simplifying the problem, looking for a pattern, using
working backwards, eliminating possibilities, using a formula or
acting out the problem, using logic, using guess and check, using a
using a computer sketching program like Geometer's Sketchpad, The
SuperSupposer, or Cabri, writing a computer program, or
a graphing calculator." (Page 71, NJMCF)
- The Truth: Problem solving in the context of
the application of remembered math
with the step-by-step use of context-specific math facts tightly linked
to context-specific math skills. It may involve remembering and
a whole series of linked facts and skills. If there are gaps in
the whole process breaks down. But New Jersey math educators think this
is too difficult for today's kids. They would rather distract New
students with thousands of hours of open-ended games and "activities",
all with the goal of "discovering" general "problem-solving skills".
say that such "skills" are all that one needs to be a "mathematical
solver". It's a cult-like belief in magic! They would have us
that it's possible to "do math" without remembering any specific math.
Connections (The third
and the third "process skill")
- "Most experiences relating to mathematical communication will
use of natural language". (Page 75, NJMCF)
- The Truth: Basic communication skills are more
skills, but they're not math. Learning about "communication" in the
of math refers to learning the precise language and symbols of math.
not about "natural language".
standard" and the fourth "process skill")
- "Mathematics must be approached as a whole" (Page 76, NJMCF)
- In a very real sense, the whole is greater than the sum of its
(Page 76, NJMCF)
- The Truth: Similar to "whole language", progressive
preach "whole math". They would have us believe that kids are ready for
complex math in the first grade. But genuine math knowledge must be
through a step-by-step buildup process, from simple to complex, with
initial hiding of complexities.
- "Reasoning" CPIs for "the end of Grade 4" say students:
The Truth: It's all unbelievable, but sadly true.
out in math are to learn that "guess and check" is the first principle
of reasoning. Next comes a weak form of inductive reasoning whereby
"generalize" based on the experience of a few tests with a calculator.
As for "explain their own thinking" and "justify answers", progressives
just want them to be happy with "their own thinking". New Jersey kids
learn about the importance of accuracy and correct answers. They will
about "socially correct thinking", where every kid's ideas are to be
as valuable, not subject to criticism. They won't learn about
correct thinking, with the quick identification and correction of
(The fifth "content standard")
- CPI 4.1: Make educated guesses and test them for correctness.
- CPI 4.2: Draw logical conclusions and make generalizations.
- CPI 4.3: Use models, known facts, properties, and
their own thinking.
- CPI 4.4: Justify answers and solution processes in a variety
Number Sense (The sixth
- "Problem Solving", "Communication", "Connections", and
also the first four NCTM "standards". But now we have something
The NJMS go beyond the NCTM Standards by elevating calculators,
and "manipulatives" to a
"standard". First they say "These tools should be used, not to replace
mental math and paper-and-pencil computational skills, but to enhance
of mathematics and the power to use mathematics". (Page 137, NJMCF) But
later they hedge, with "Students must understand how to add, subtract,
multiply, and divide whole numbers, fractions, and other kinds of
With calculators that perform these operations quickly and accurately,
however, the instructional emphasis now should be on understanding the
meaning and uses of operations, and on estimation and mental skills,
than solely on developing paper-and-pencil skills." (Page 251, NJMCF)
- The Truth: Don't be fooled by the hedging language.
will occur because New Jersey math educators believe it should. They
that every child must have a calculator, beginning in kindergarten, and
they leave it up to the kids to "choose" the "appropriate"
method. Of course, any child will opt for the simplicity of calculator
button-pushing, when faced with the more difficult alternative of
basic math facts and acquiring paper-and-pencil skills. But progressive
educators aren't bothered by this. They say that traditional
skills are now obsolete.
- The result? If easily-acquired calculator skills are
more-difficult, step-by-step buildup of remembered number manipulation
skills, New Jersey students will never be able to build more
math knowledge in the brain. They will be missing the key, first step
growing intellectually in the ways uniquely conveyed by learning
Geometry and Spatial Sense (The seventh "content
- "A "sense-building mode" is best established when students are
with opportunities to explore number relationships, are encouraged to
and challenge, and are allowed to discover strategies and techniques of
their own that ease the path to the solution of mathematical problems."
(Page 173, NJMCF)
- The Truth: Math isn't the place for
methods. Each child needs to learn the standard methods of traditional
K-12 math. Otherwise, Sarah may "discover" that 2 x 2 = 22, using "her
own techniques", and progressive teachers won't want to tell her she's
Numerical Operations (The eighth "content standard")
- "Traditionally, elementary school geometry instruction has
characterization of shapes; at the secondary level, it has been taught
as the prime example of a formal deductive system. While these
of the content are important, they are also limiting. In order to
spatial sense, students should be exposed to a broader range of
activities at all grade levels". (Page 209, NJMCF)
- The Truth: Perhaps you interpreted the preceding
as supporting both the "traditional" geometry topics and the new
range"? With all the extra time made available by the use of
why not? But no! Progressive educators want to replace traditional math
content with vague, open-ended discovery learning. In particular, they
reject plane geometry, because it requires careful, disciplined
and it can't be done by a calculator. They think it's too hard for New
Measurement (The ninth "content standard")
- Initially, the NJMCF sounds encouragingly traditional,
must know the basic facts of the multiplication table" (Page 251,
and ""Most paper-and-pencil procedures should still be taught
one-digit basic facts should still be committed to memory"
252, NJMCF) (emphasis in the original). But then the progressive
take over. They say "Students should be able to quickly and
easily recall one-digit sums and differences. The most effective way to
accomplish this has been shown to be the focused and explicit use of
fact strategies". The "strategies"? They're called "counting on",
back", "make ten", "doubles", and "near doubles". (Page 254, NJMCF)
- The Truth: Today's progressive educators detest the
basic number facts. They would prefer to have kids learn an array of
for "constructing" the sum or difference each and every time.
the point? Why not memorize the addition and multiplication tables,
the mind to learn more advanced math? The result of these "figure it
each time" methods will be kids who need a calculator to multiply 7
Estimation (The tenth
- "Estimation of measures should be the focus of any work that
with measurement. Indeed, the very concept that any continuous
is inexact -- that it is at best an "estimate" -- is a concept that
be developed throughout the grades." (Page, 281, NJMCF)
- The Truth: The historical development of
was primarily motivated by the search for a general method for
the exact area of any region in the plane which is bounded by a
perimeter. Perhaps New Jersey math educators haven't heard. The search
was successful. Integral calculus is routinely applied, thousands of
daily, to exactly measure areas and volumes. Will New Jersey high
graduates go away thinking that all "measurement is inexact"?
and Functions (The eleventh "content standard")
- "One of the estimation emphases for very young children is the
of the idea that guessing is an important and exciting part of
(Page 311, NJMCF)
- "People who use mathematics in their lives and careers find
to be preferable to the use of exact numbers in many circumstances."
- The Truth: Since progressives reject the idea of
the traditional procedures of exact math, they are left with no
but to elevate guessing and estimating, and trash exactness. This
will result in kids who can't find exact answers. More alarmingly, New
Jersey students won't see the importance of exact answers. Do you want
them building your bridges?
Probability and Statistics (The twelfth "content
- "From the earliest age, students should be encouraged to
patterns that they find in numbers, shapes, and expressions, and by
so, make mathematical discoveries." (Page 335, NJMCF)
- The Truth: The pattern-recognition learning
the NJMCF amount to aimless playing, not the carefully focused,
buildup of math knowledge.
- "Students need to have experience with situations involving
polynomial, trigonometric, exponential, and rational functions" (Page
- The Truth: The "experience" they are talking about
consists of "activities"
using graphing calculators. With or without one, New Jersey high school
graduates will be encouraged to use "guess and check" and estimation
solving any equation.
- "From weather reports to sophisticated studies of genetics,
results to product preference surveys, probability and statistical
and concepts are increasingly present in the media and in everyday
Students need this mathematics to help them judge the correctness of an
argument supported by seemingly persuasive data." (Page 371, NJMCF)
- "They should not believe an argument merely because various
are offered. Rather, they should be able to judge whether the
are meaningful and are being used appropriately." (Page 371, NJMCF)
- The Truth: The preceding two statements are
totally unrealistic expectations being sold by progressive educators.
would have kids believe that they are savvy experts because they have
"the theoretical probabilities of different events such as the possible
sums of a pair of dice." (Page 371, NJMCF)
Discrete Mathematics (The fourteenth "content standard")
- "Modern technology provides tools for supplementing the
on algebraic techniques, such as solving equations, with a more visual
perspective, with graphs of equations displayed on a screen." (Page
- The Truth: This isn't about "supplementing"
It's about replacing it with calculators. Consider CPI 13.11: "Explore
linear equations through the use of calculators, computers, and other
Yes, without a graphing calculator, New Jersey kids won't be able to
equation of a straight line.
Conceptual Building Blocks of Calculus (The fifteenth
- "The material in this chapter is drawn from activities that
and classroom-tested by the K-12 teachers in the Rutgers University
Program in Discrete Mathematics over the past nine years; this program
is funded by the National Science Foundation." (Page 442, NJMCF)
- The Truth: Sounds impressive, but, sad to say, it was
of time and money. All these NJMCF "activities" are based on the
misguided belief that each kid can personally discover the major facts
of K-12 math, without direct instruction from teachers or books.
the key ideas will all emerge from hundreds of activities. For example,
let's take the key idea of counting formulas for permutations and
and consider how they're handled in the 46 pages of Discrete Math
in the NJMCF. Yes, the words "permutation" and "combination" are
and there is an "activity" (Page 481, NJMCF) which suggests that kids
discover these formulas based on generalizing from one case. But if
looking for any suggestion of unusual importance, or expect to see the
derivation, explanation, or application of these formulas, don't
It's expected that each kid will somehow recognize the importance as
rediscover key ideas that countless mathematicians have carefully
over hundreds of years. If this underlined clause sound like an
consider the following quote from page 28 of the 1992 "research" paper,
A Constructivist Alternative to the Representational View of
Mathematical Education, (Journal for Research in Mathematics
23, 2-33). These progressive gurus actually said "it is possible for
students to construct for themselves the mathematical practices that,
took several thousands of years to evolve". Simply amazing!
Excellence and Equity for All Students (The sixteenth
- "An intuitive feel for the mathematics of infinity, limit, and
accessible and necessary for all students. Although some students will
go on to study these concepts in a formal calculus class, this standard
does not advocate the formal study of calculus in high school for all
or even for all college-intending students. Rather, it calls for
opportunities for all students to informally investigate the central
of calculus." (Page 487, NJMCF)
- The Truth: Do you sense the fear? Apparently genuine
too difficult for most New Jersey high school students, so New Jersey
have decided to substitute their own "informal" calculus. Parents of
Jersey, don't subject your children to this nonsense. Let them wait for
genuine calculus. Otherwise, they will have to eventually unlearn the
feel" proposed here.
- The Truth: With the proper, grade-by-grade buildup of
during the K-10 years, the average student could begin to learn genuine
calculus, beginning in the third year of the high school.
- "The first fifteen standards set out high expectations for all
and this final standard insists that all students need to, can,
and will meet those standards." (Emphasis in the original)
- The Truth: The good news is that New Jersey kids won't
difficulty meeting these so-called "standards", because there are no
expectations. The bad news is that New Jersey kids will "graduate" from
high school with the math education that they will pick up from their
supplemented by the bits-and-pieces they manage to "discover" under the
heading of "math" during the K-12 years.
William G. Quirk, Ph.D.