The Truth About The New Jersey
Math Standards
An Analysis of the New Jersey Math Standards (NJMS)
by Bill Quirk (E-Mail:
wgquirk@wgquirk.com)
The New Jersey Math Standards Are
Based On the NCTM Standards
The New
Jersey Math Standards (NJMS) are a "child" of The NCTM
Standards, from The National Council of Teachers of Mathematics
(NCTM).
Similar to the parent, they fail to identify the specific,
grade-by-grade
math content that should be taught and learned during the K-12 years.
Instead,
they promote the NCTM's "progressive" program for replacing traditional
K-12 math with "manipulatives", calculators, "guess and check",
estimation,
and content-independent "process skills". These NJMS substitution
strategies
are disguised as 16 so-called "content standards", each with multiple
"cumulative
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
solution
to 3x + 2 = 17, and they will be able to use a calculator to estimate
the
solution to 6x +2 = 22, but don't be surprised if they can't find the
exact
answer in the second case. Expect to hear that "exact answers" aren't
important
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
selling
price of an $8 item, now offered at a 25% discount.
The NJMS achieves its lean 26-page appearance by speaking in
generalities
and by suppressing the underlying "progressive" philosophy found in the
NCTM
Standards. That's all found in the 670-page New
Jersey Mathematics Curriculum Framework (NJMCF), a 1500
"activities"
cookbook for math self-discovery. Following the lead of the NCTM, the
following
"progressive" axioms have been assumed by the writers of the NJMCF:
- Belief that children must be allowed to follow their own
interests to
personally discover the math knowledge that they find interesting and
relevant
to their own lives.
- Rejection of the concept of a common core of basic math
knowledge that
all children should learn during the K-12 years
- Belief that children must "construct" mathematical knowledge for
themselves.
- Rejection of teacher-directed knowledge transmission.
- Belief that all knowledge must be acquired as a byproduct of
social
interaction
in real-world settings.
- Rejection of classroom learning.
- Belief in the primary importance of general, content-independent
"process"
skills.
- Rejection of the need to remember any specific math content.
- Belief that calculators have fundamentally changed the nature of
math.
- Rejection of the need to acquire traditional paper-and-pencil
math
skills.
- Belief that learning must always be an enjoyable, happy
experience,
with
knowledge emerging naturally from games and group activities.
- Rejection of any attempt to challenge a child to work harder.
When it comes to the learning math, progressives preach their gospel of
"discovering math through problem solving". You may think this
refers
to the traditional process whereby teachers ask questions and present
problems
which have been carefully chosen to lead students to discover
teacher-targeted
math knowledge. Not so! Progressives preach open-ended "exploration",
with
no expectation that different kids will "discover" the same thing.
Forget
about a careful step-by-step buildup of core math knowledge that all
students
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
attempts
to solve complex, real-world problems, with the further complexity that
the problems must be chosen by the students, based on their personal
interests.
Progressives don't believe it's right to pre-specify what kids should
learn,
and they don't believe that all kids should be required to learn the
same
content. This in turn forces them to redefine the meaning of
"testing"
to equate it with "finding out what each kid has discovered", rather
than
identifying what each student has failed to learn. More generally, New
Jersey math educationists have redefined the meaning of
"standards",
"math", and "learning math".
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
learning",
"holistic learning", "developmentally appropriate", and
"research-based",
you'll know you're experiencing a strange encounter with a member of
the
progressive educationist cult. Good luck!
The good news is that genuine researchers are also appalled by the
claims
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
Professor
Hirsch discussed the difficulty of publishing an important research
article,
Applications
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.
Reder,
have the highest respect of those who do genuine research in cognitive
psychology. Most progressive educationists don't know about these
scholars
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
with
the linked sections from The Truth About the
NCTM
Standards. This will be more than enough for you to fully
appreciate
how New Jersey educationists want to emphasize "progressive" social
goals
and the fundamental progressive gospel of "discovering math through
problem
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
Standards
The NJMS
fail to qualify as genuine math standards when evaluated according to
the
Characteristics
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
NJMCF
fail to do this. They are:
- Not focused
- The 16 NJMS "content standards" are very general learning
objectives,
not
focused math topics. Each NJMS "standard" has an associated list of
"Cumulative
Progress Indicators" (CPIs) which still fail to identify focused math
topics.
But this is the lowest level of detail found in the NJMS! Consider
the
very general nature of the following CPIs, listed for the end of
grade
4:
(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
standard
N.)
- CPI 1.1: Use discovery-oriented, inquiry-based, and
problem-centered
approaches
to investigate and understand mathematical content appropriate to early
elementary grades.
- CPI 1.10 substitutes "content appropriate to the middle
grades", and
CPI
1.15 substitutes "content appropriate to the high school grades".
- The Truth: The detailed identification of
"mathematical content
appropriate" for each grade is exactly the role of math standards!
- CPI 2.2: Identify and explain key mathematical concepts, and
model
situations
using oral, written, concrete, pictorial, and graphical methods.
- CPI 3.1: View mathematics as an integrated whole rather than
as a
series
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,
and other
tools based on their utility and limitations and on the problem
situation.
- CPI 6.1 Use real-life experiences, physical materials, and
technology
to
construct meaning for whole numbers, commonly used fractions, and
decimals.
- CPI 7.5 Investigate and predict the results of combining,
subdividing,
and changing shapes.
- CPI 8.5: Use a variety of mental computation and
estimation
techniques
- CPI 10.4: Explore, construct, and use a variety of estimation
strategies.
- CPI 11.5: Observe and recognize examples of patterns,
relationships,
and
functions in other disciplines and contexts.
- CPI 12.7: Make predictions based on intuitive, experimental,
and
theoretical
probabilities.
- The NJMCF goes one level below the CPI, associating multiple
activities
and games with each CPI. But there's still an overall failure to
identify
focused math topics, and there's frequently no evident reason as to why
a particular activity or game is associated with a particular CPI.
Somehow,
the NJMCF "activities" are supposed to help New Jersey kids discover
some
math. What math? That's usually not clear.
- Not specific
- New Jersey educationists conspicuously avoid being specific
about math
content for the K-12 years. They believe that each kid should be
allowed
to explore broadly.
- Not basic
- New Jersey math educators don't believe in identifying and
teaching
core
math knowledge. Instead, they endorse broad, discovery
learning (Link to Chapter 4 of The Truth About The NCTM Standards)
- Not teachable
- The NJMS and NJMCF are too broad and unfocused to be teachable.
But
"teachable"
isn't a term that has meaning for progressives. Their endorsement of
discovery
learning implies a corresponding rejection of teacher-directed
knowledge
transmission. "Progressive" educators believe it is wrong for teachers,
or anyone else, to decide what kids should learn. They say the main
role
of the teacher is to create
rich enabling environments (Link to Chapter 4 of The Truth About
The
NCTM Standards)
- Not measurable
- Because of their fundamental belief in "broad content" and
"discovery
learning",
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
success". (Link
to Chapter 4 of The Truth About The NCTM Standards)
- Not linked to grade
- The NJMS and NJMCF are not specific about what should happen in
each
grade.
They discuss general learning goals for the ends of grade 4, 8, and 12.
There is no possibility for either student or teacher accountability.
- Concise, to a fault
- CPI's are concisely stated. That is, they do "express much in
few
words".
The problem is that they express far too much, using general words.
- Redundant
- Consider the redundancy in the following examples:
- CPI 8.5: Use a variety of mental computation and estimation
techniques.
- CPI 8.6: Select and use appropriate computational methods
from mental
math,
estimation, paper-and-pencil, and calculator methods and check for
reasonableness
of results.
- CPI 8.10: Develop, apply, and explain procedures for
computation and
estimation
with whole numbers, fractions, decimals, integers, and rational numbers.
- CPI 10.4: Explore, construct, and use a variety of estimation
strategies.
- CPI 10.6: Determine the reasonableness of an answer by
estimating the
result
of operations.
- CPI 10.7: Apply estimation in working with quantities,
measurement,
time,
computation, and problem solving.
- CPI 10.8: Develop, apply, and explain a variety of different
estimation
strategies involving quantities and measurement.
- Not genuine math
- This is the most alarming problem and the topic of the
next
section.
- Not brief
- Relative to the 217 page NCTM parent, the 26 page NJMS is
brief. But
the
fair comparison is to include the 670-page NJMCF, for a total 696 pages.
- Not selective
- 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.
- Not pedagogically neutral
- The NJMS don't explicitly preach "progressive" teaching
methods. But
they
assume them! All the key words are there. The detailed progressive
rational
and "teaching" methods are found in the 670-page NJMCF
The NJMS Offer A Program For
Replacing
Traditional K-12 Math With Calculators, Math Appreciation, and
Content-Independent
"Process Skills"
Chapter 3 and chapter 4
of The Truth About The NCTM Standards
explains
how the NCTM wants to replace traditional core math knowledge with
calculators,
math appreciation, and general, content-independent "process skills".
These
"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
first
"content
standard" and the first "process skill")
- The NJMCF say that the problem-solving "process skills" to be
learned
are
"making a list, making a chart or a table, drawing a diagram, making a
model, simplifying the problem, looking for a pattern, using
manipulatives,
working backwards, eliminating possibilities, using a formula or
equation,
acting out the problem, using logic, using guess and check, using a
spreadsheet,
using a computer sketching program like Geometer's Sketchpad, The
Geometry
SuperSupposer, or Cabri, writing a computer program, or
using
a graphing calculator." (Page 71, NJMCF)
- The Truth: Problem solving in the context of
math
refers to
the application of remembered math
knowledge,
with the step-by-step use of context-specific math facts tightly linked
to context-specific math skills. It may involve remembering and
applying
a whole series of linked facts and skills. If there are gaps in
knowledge,
the whole process breaks down. But New Jersey math educators think this
is too difficult for today's kids. They would rather distract New
Jersey
students with thousands of hours of open-ended games and "activities",
all with the goal of "discovering" general "problem-solving skills".
They
say that such "skills" are all that one needs to be a "mathematical
problem
solver". It's a cult-like belief in magic! They would have us
believe
that it's possible to "do math" without remembering any specific math.
- Communication (The
second
"content standard" and the second "process skill")
- "Most experiences relating to mathematical communication will
involve
the
use of natural language". (Page 75, NJMCF)
- The Truth: Basic communication skills are more
important
than math
skills, but they're not math. Learning about "communication" in the
context
of math refers to learning the precise language and symbols of math.
It's
not about "natural language".
- Connections (The third
"content standard"
and the third "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
parts."
(Page 76, NJMCF)
- The Truth: Similar to "whole language", progressive
educationists
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
acquired
through a step-by-step buildup process, from simple to complex, with
the
initial hiding of complexities.
- Reasoning (The
fourth "content
standard" and the fourth "process skill")
- "Reasoning" CPIs for "the end of Grade 4" say students:
- 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
relationships to
explain
their own thinking.
- CPI 4.4: Justify answers and solution processes in a variety
of
problems.
- The Truth: It's all unbelievable, but sadly true.
Kids
just starting
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
kids
"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
won't
learn about the importance of accuracy and correct answers. They will
learn
about "socially correct thinking", where every kid's ideas are to be
seen
as valuable, not subject to criticism. They won't learn about
mathematically
correct thinking, with the quick identification and correction of
errors
in understanding.
- Tools and
Technology
(The fifth "content standard")
- "Problem Solving", "Communication", "Connections", and
"Reasoning", are
also the first four NCTM "standards". But now we have something
different!
The NJMS go beyond the NCTM Standards by elevating calculators,
computers,
and "manipulatives" to a
full-fledged
"standard". First they say "These tools should be used, not to replace
mental math and paper-and-pencil computational skills, but to enhance
understanding
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
numbers.
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,
rather
than solely on developing paper-and-pencil skills." (Page 251, NJMCF)
- The Truth: Don't be fooled by the hedging language.
Calculator "replacement"
will occur because New Jersey math educators believe it should. They
insist
that every child must have a calculator, beginning in kindergarten, and
they leave it up to the kids to "choose" the "appropriate"
computational
method. Of course, any child will opt for the simplicity of calculator
button-pushing, when faced with the more difficult alternative of
memorizing
basic math facts and acquiring paper-and-pencil skills. But progressive
educators aren't bothered by this. They say that traditional
computational
skills are now obsolete.
- The result? If easily-acquired calculator skills are
substituted for
the
more-difficult, step-by-step buildup of remembered number manipulation
skills, New Jersey students will never be able to build more
sophisticated
math knowledge in the brain. They will be missing the key, first step
towards
growing intellectually in the ways uniquely conveyed by learning
genuine
math.
- Number Sense (The sixth
"content
standard")
- "A "sense-building mode" is best established when students are
provided
with opportunities to explore number relationships, are encouraged to
question
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
self-discovered,
personal
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
wrong.
- Geometry and Spatial Sense (The seventh "content
standard")
- "Traditionally, elementary school geometry instruction has
focused on
the
characterization of shapes; at the secondary level, it has been taught
as the prime example of a formal deductive system. While these
perspectives
of the content are important, they are also limiting. In order to
develop
spatial sense, students should be exposed to a broader range of
geometric
activities at all grade levels". (Page 209, NJMCF)
- The Truth: Perhaps you interpreted the preceding
statement
as supporting both the "traditional" geometry topics and the new
"broader
range"? With all the extra time made available by the use of
calculators,
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
thinking,
and it can't be done by a calculator. They think it's too hard for New
Jersey kids.
- Numerical Operations (The eighth "content standard")
- Initially, the NJMCF sounds encouragingly traditional,
with "Fourth-graders
must know the basic facts of the multiplication table" (Page 251,
NJMCF)
and ""Most paper-and-pencil procedures should still be taught
and
one-digit basic facts should still be committed to memory"
(Page
252, NJMCF) (emphasis in the original). But then the progressive
writers
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
basic
fact strategies". The "strategies"? They're called "counting on",
"counting
back", "make ten", "doubles", and "near doubles". (Page 254, NJMCF)
- The Truth: Today's progressive educators detest the
idea
of memorizing
basic number facts. They would prefer to have kids learn an array of
"techniques"
for "constructing" the sum or difference each and every time.
What's
the point? Why not memorize the addition and multiplication tables,
freeing
the mind to learn more advanced math? The result of these "figure it
out
each time" methods will be kids who need a calculator to multiply 7
times
9.
- Measurement (The ninth "content standard")
- "Estimation of measures should be the focus of any work that
students
do
with measurement. Indeed, the very concept that any continuous
measurement
is inexact -- that it is at best an "estimate" -- is a concept that
must
be developed throughout the grades." (Page, 281, NJMCF)
- The Truth: The historical development of
integral
calculus
was primarily motivated by the search for a general method for
measuring
the exact area of any region in the plane which is bounded by a
continuous
perimeter. Perhaps New Jersey math educators haven't heard. The search
was successful. Integral calculus is routinely applied, thousands of
times
daily, to exactly measure areas and volumes. Will New Jersey high
school
graduates go away thinking that all "measurement is inexact"?
- Estimation (The tenth
"content
standard")
- "One of the estimation emphases for very young children is the
development
of the idea that guessing is an important and exciting part of
mathematics."
(Page 311, NJMCF)
- "People who use mathematics in their lives and careers find
estimation
to be preferable to the use of exact numbers in many circumstances."
(Page
309, NJMCF)
- The Truth: Since progressives reject the idea of
learning
the traditional procedures of exact math, they are left with no
alternative
but to elevate guessing and estimating, and trash exactness. This
program
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?
- Patterns,
Relationships,
and Functions (The eleventh "content standard")
- "From the earliest age, students should be encouraged to
investigate
the
patterns that they find in numbers, shapes, and expressions, and by
doing
so, make mathematical discoveries." (Page 335, NJMCF)
- The Truth: The pattern-recognition learning
activities
found in
the NJMCF amount to aimless playing, not the carefully focused,
step-by-step
buildup of math knowledge.
- "Students need to have experience with situations involving
linear,
quadratic,
polynomial, trigonometric, exponential, and rational functions" (Page
365,
NJMCF)
- 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
for
solving any equation.
- Probability and Statistics (The twelfth "content
standard")
- "From weather reports to sophisticated studies of genetics,
from
election
results to product preference surveys, probability and statistical
language
and concepts are increasingly present in the media and in everyday
conversations.
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
statistics
are offered. Rather, they should be able to judge whether the
statistics
are meaningful and are being used appropriately." (Page 371, NJMCF)
- The Truth: The preceding two statements are
typical
of the
totally unrealistic expectations being sold by progressive educators.
They
would have kids believe that they are savvy experts because they have
discussed
"the theoretical probabilities of different events such as the possible
sums of a pair of dice." (Page 371, NJMCF)
- Algebra (The
thirteenth
"content standard")
- "Modern technology provides tools for supplementing the
traditional
focus
on algebraic techniques, such as solving equations, with a more visual
perspective, with graphs of equations displayed on a screen." (Page
405,
NJMCF)
- The Truth: This isn't about "supplementing"
traditional algebra.
It's about replacing it with calculators. Consider CPI 13.11: "Explore
linear equations through the use of calculators, computers, and other
technology."
Yes, without a graphing calculator, New Jersey kids won't be able to
find the
equation of a straight line.
- Discrete Mathematics (The fourteenth "content standard")
- "The material in this chapter is drawn from activities that
have been
reviewed
and classroom-tested by the K-12 teachers in the Rutgers University
Leadership
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
all
a waste
of time and money. All these NJMCF "activities" are based on the
terribly
misguided belief that each kid can personally discover the major facts
of K-12 math, without direct instruction from teachers or books.
Somehow
the key ideas will all emerge from hundreds of activities. For example,
let's take the key idea of counting formulas for permutations and
combinations,
and consider how they're handled in the 46 pages of Discrete Math
"activities"
in the NJMCF. Yes, the words "permutation" and "combination" are
mentioned,
and there is an "activity" (Page 481, NJMCF) which suggests that kids
will
discover these formulas based on generalizing from one case. But if
you're
looking for any suggestion of unusual importance, or expect to see the
derivation, explanation, or application of these formulas, don't
bother!
It's expected that each kid will somehow recognize the importance as
they personally
rediscover key ideas that countless mathematicians have carefully
refined
over hundreds of years. If this underlined clause sound like an
exaggeration,
consider the following quote from page 28 of the 1992 "research" paper,
A Constructivist Alternative to the Representational View of
Mind in
Mathematical Education, (Journal for Research in Mathematics
Education,
23, 2-33). These progressive gurus actually said "it is possible for
students to construct for themselves the mathematical practices that,
historically,
took several thousands of years to evolve". Simply amazing!
- Conceptual Building Blocks of Calculus (The fifteenth
"content standard")
- "An intuitive feel for the mathematics of infinity, limit, and
change
are
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
students
or even for all college-intending students. Rather, it calls for
providing
opportunities for all students to informally investigate the central
ideas
of calculus." (Page 487, NJMCF)
- The Truth: Do you sense the fear? Apparently genuine
calculus is
too difficult for most New Jersey high school students, so New Jersey
educationists
have decided to substitute their own "informal" calculus. Parents of
New
Jersey, don't subject your children to this nonsense. Let them wait for
genuine calculus. Otherwise, they will have to eventually unlearn the
"intuitive
feel" proposed here.
- The Truth: With the proper, grade-by-grade buildup of
math
knowledge
during the K-10 years, the average student could begin to learn genuine
calculus, beginning in the third year of the high school.
- Excellence and Equity for All Students (The sixteenth
and
last "content
standard")
- "The first fifteen standards set out high expectations for all
students,
and this final standard insists that all students need to, can,
and will meet those standards." (Emphasis in the original)
(Page
517, NJMCF).
- The Truth: The good news is that New Jersey kids won't
have any
difficulty meeting these so-called "standards", because there are no
measurable
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
parents,
supplemented by the bits-and-pieces they manage to "discover" under the
heading of "math" during the K-12 years.
Copyright
1997-2011
William G. Quirk, Ph.D.