A Summary View of NCEE Math
Includes the New York City
Modifications
NCEE
Math is Based on the NCTM Standards
We use the term "NCEE math" to refer to
the
"math" content and skills demonstrated in the New
Standards™ Performance Standards (NSPS), a 3volume product
of
the National Center on Education and the Economy (NCEE) and the
University of Pittsburgh. Recently renamed as the America's
Choice™ Performance Standards (ACPS), these materials include a
total
of 38 math "Work Sample & Commentary" sets (WS&C sets), with 19
for elementary school, 12 for middle school, and 7 for high
school.
Each WS&C set consists of one or more NCEE math tasks, samples of
student
methods for carrying out each task, and extensive NCEE comments. Chapter
2 (elementary school),
Chapter
3 (middle school), and Chapter
4 (high school) provide a detailed analysis of all 38 NCEE WS&C
sets.
The NCEE says their math performance
standards
are "based directly" on the "content standards" developed by the
National
Council of Teachers of Mathematics (NCTM). Claiming that the NCTM
standards specify "what students should know and be able to do,"
the NCEE says that their performance standards "go the next step" by
illustrating
"good tasks" and "how good is good enough."
What's the next step after the
NSPS?
That's the New Standards Reference Exam (NSRE). Recently
renamed
as the America's Choice Reference Exam (ACRE), this NCEE
product
is used statewide in Rhode Island and Vermont. More generally,
the
NCEE claims that their products are now used in more than 33 states.
New York City has developed their own
version
of the NSPS. Interestingly, the NYC authors
claim
that their student work samples come "from schools throughout the
city."
But 33 of the 38 NYC WS&C sets also appear with the same work
samples
in the NSPS, and the NCEE claims input from a "diverse range of
students
in a wide variety of settings." New York City is mentioned
as one source, but eight other specific locations are also credited.
This report covers the NYC
modifications.
In brief, the NYC authors modified the NSPS by deleting 5 NSPS sets,
adding
5 NYC (only) sets, and changing some student work samples for 6 NSPS
sets.
As the details in later chapters reveal, these modifications don't
justify
the born in NYC claim.
NCEE Math is
"World Class,"
According to International Experts
The NCEE says their performance standards"not
only provide clear expectations for student achievement, but also
include
numerous examples of student work that show what work that meets
standards
looks like." They
claim to be offering "world
class
standards" that "have
been benchmarked to the expectations of those countries with the
highest
student performance in the world."
They further claim that their product has been reviewed by researchers
and recognized experts in several other countries, including Germany,
Japan,
and Singapore. They say that "no
reviewer identified a case of significant omission," and "none of the
reviewers
identified standards for which the expectations expressed in the
standards
were less demanding than those for students in other countries."These
claims are found on pages 4 and 5 in all three NSPS volumes.
The evidence provided here shows that
mathematicians
were not included in the experts consulted. Mathematicians aren't
impressed with "math content" found in the NCTM standards, and no
mathematician
would judge the NCEE math performance standards to be an acceptable
guide
to the math knowledge that should be acquired during the K12 years.
Similar to the NCTM standards, the NSPS
is a product of constructivist math educators who believe that every
child
is entitled to know as much math as every other child. Although
they
don't shout it from the rooftops, they also believe that genuine math
is
the domain of "privileged" white males and too difficult for women,
minorities,
and the poor.
In the first major section of this
chapter
you will learn how the NCEE severely limits K12 math content.
This
is the key strategy for achieving a social agenda that emphasizes
equality
of results and the current happiness of the child.
Without
a solid base of remembered math content knowledge, genuine math problem
solving, reasoning, and conceptual understanding aren't
possible.
In the second major section of this chapter you will learn how the NCEE
gets around this by emphasizing contentindependent skills.
Note: Generally speaking,
the discussion in this chapter applies to both the NSPS and the NYC
version
of the NSPS. Exceptions will be given in context.
Shallow
Math Content
Chapters 2, 3, and 4 cover the NCEE math
WS&C
sets in the sequence found in the NSPS source materials. In this
section we group the WS&C math tasks by content area.
This
organization more clearly reveals the minimal content expectations
found
in NCEE math.
How the NCEE
Limits
Arithmetic
Calculator were used for all NCEE middle
school
and high school computational tasks. Although the full
computational
details are never made clear, the NCEE doesn't mention calculator use
for
elementary school computations. Here's the complete list of NCEE
K5 computational tasks:
 Share 25 objects as "equally as
possible"
among four friends.
 The objects are 25 balloons, 25
dollars, and
25 cookies.
 Calculate 63 x 46.
 Calculate 522  367 in two
different
ways.
 Calculate 87 x 9 in two different
ways.
 Demonstrate multiple ways to
calculate
62
x 85.
 Share 3 bags of M & M's
equally
among
four children. Each bag contains 52 M&M's.
 Calculate 13 x 14.
There are no NCEE K12 math tasks
involving
large numbers, negative numbers, prime numbers, operations with
fractions,
or operations with decimals. Following the lead of the NCTM, the
NCEE points to the power of calculators. They believe that
today's
students only need experience with small "familiar numbers" and an
understanding
of "the meaning" of operations.
For a comprehensive list of the missing
arithmetic topics, please see the Number Sense sections for
kindergarten
through grade 7 in the California Mathematics Content Standards,
Chapter
2 of The California
Mathematics
Framework
How the NCEE
Limits
Algebra and Functions
The NCEE classifies the following tasks
under
"Functions and Algebra:"
 Elementary School Functions and
Algebra
Tasks
 Give examples of "linear" and
"nonlinear"
patterns. ["Linear" is defined as "repeating," and "nonlinear"
is
defined as "grow or lower itself."]
 Middle School Functions and
Algebra
Tasks
 "How many segments are
needed to
connect
5 points? 6 points? 8 points? 10 points? 30
points?
100 points? n points?"
 Find the equation of the
straight
line connecting
the two points.
 High School Functions and
Algebra
Tasks
 Given the formula D = S x
T,
solve for
T when given values for S and D.
 Given the formula N = R x
T,
solve for
R when given values for N and T.
 Find a formula for the number of
hidden small
cubes in a large cube consisting of n^{3} small cubes.
 Find a formula for the number of
visible
small cubes in a large cube consisting of n^{3} small cubes.
 If 96 cm shopping carts
are
stored in
a stack, and each new cart adds 28.8 cm to the length of the nested
stack,
find a formula that expresses S, the total length of the stack,
in
terms of N, the total number of carts in the stack. Also
find
a formula that expresses N in terms of S.
 A rectangular block of cheese of
volume V
is cut into identical small cubes. One layer of the small cubes
exactly
fills a rectangular pan of area A. Given that the
side
of a small cube is of length L. Find L in terms of V and
A.
Find N, the number of small cubes, in terms of V and A.
 If an item originally sold for
$120,
and the
store offers a 50% discount at the end of each week that the item
remains
unsold, find the selling price of the item at the end of the second
week.
[NYC only]
 Given the monthly base fee and
charge per
call for two local phone service plans, compare the plans for several
cases.
[NYC only]
For the visible small cubes formula (high
school task #4), the student added n^{2}, n^{2}
 n, and n^{2} 2n + 1 to get the sum 3n^{2}  3n +
1.
That's as difficult as it gets in NCEE algebra. Symbolic
manipulation
is largely missing, with nothing about factoring, simplifying, or
quadratic
equations. NCEE functions are limited to linear functions.
For a comprehensive list the
missing
algebra and functions topics, please see the California Mathematics
Content
Standards, Chapter 2 of The
California Mathematics Framework. In particular, see the the
algebra and functions sections for kindergarten through grade 7,
and also see the algebra I, algebra II, and linear algebra sections for
high school.
There are two middle school tasks that
offer the opportunity for genuine algebra, but the NCEE didn't classify
either under "Functions and Algebra".
 "A length of string that is 180 cm
long is
cut into 3 pieces. The second piece is 25% longer than the first
and the third piece is 25% shorter than the first. How long is
each
piece?"
 Given 7 equations involving 10
variables,
where each variable represents a unique digit between 0 and 9, find the
value of all 10 variables.
The Equations:
 g + g + g = d
 j + e = j
 g^{2} = d
 b + g = d
 f  b = c
 i/h = a (h > a)
 a x c = a
For the first task, the NCEE authors
appear pleased that the only student method offered "does not show a
standard
approach that a student of algebra might take." The method
demonstrated
is a guessing strategy that only works with carefully chosen
numbers.
For the second task, the student solved the system of equations
by
using substitution, properties of equality, and knowledge of identity
elements.
This qualifies as genuine algebra. It's the only
demonstration
of these skills found in NCEE math. Unfortunately, because of the
"unique digit between 0 and 9" constraint, this system of equations can
be easily solved using "guess and check."
Note: The New York City
authors did classify both these tasks under "Functions and
Algebra."
They also improved on the NSPS source by offering a genuine algebra
solution
for the first task. Good! But it must also be
pointed
that all other NYC improvements are simply "wise
omissions."
How the NCEE Limits
Geometry
Here we list the geometry tasks found in
the
38 WS&C sets.
 Elementary School Geometry Tasks
 Given pictures of 3D "cube
buildings," draw
the front, side, and top views.
 Draw a pattern for making a
cube. [NYC
only]
 Estimate and then measure the
"circumference"
and "diameter" of a pumpkin.
 Suppose you are given a string
that
is sixteen
inches long. If you cut or fold it in any two places, will it
always
make a triangle?"
 Use tangram manipulatives to
show
that a given
"small tangram parallelogram" is equivalent to one out of four equal
parts
of a given "large tangram parallelogram."
 Using "pattern blocks," find the
size of six
geometric shapes relative to the size of a given shape.
 Given the length and width of a
rectangle,
find the area of a rectangle
 Select a polyhedron from choices
on
a poster.
Use a compass, rubber bands, etc. to build the polyhedron of your
choice.
 Middle School Geometry Tasks
 Given the area of a circle, find
the
radius
of the circle.
 Given a 3 cm length for the edge
of
a cube,
find the volume and surface area of the cube.
 Test Eddies' claim: the surface
area
of a
cube is always twice the volume of the cube.
 Build a 1 to 10 scale model by
enlarging or
shrinking an everyday object.
 High School Geometry Tasks
 Find the altitude and area of a
triangle.
 Use properties of isosceles and
right triangles.
 Apply the Pythagorean Theorem.
 Find the area of a sector of a
circle.
 Apply the formula for the
circumference of
a circle.
 Find the volume
of a
rectangular
prism. [NYC only]
The term "theorem" is found in NCEE
geometry,
but it's always preceded by "Pythagorean." You won't find the
term
"transversal" or genuine high school geometry. For a
comprehensive
list of the missing geometry topics, please see the California
Mathematics
Content Standards, Chapter 2 of The
California Mathematics Framework. In particular, see
the
the measurement and geometry sections for kindergarten through grade 7,
and see the geometry section for high school.
How the NCEE
Limits
Statistics, Data Analysis, and Probability
The NCEE classifies the following tasks
under
"Statistics and Probability:"
 Elementary School Statistics
and
Probability
Tasks
 Compare eight pumpkins for
weight,
diameter,
circumference, height, and number of seeds.
 Analyze 24 student
heights.
Find the
range, median, and identify outliers. Develop a line plot.
 List all possible combinations
for
two dice,
list all possible sums for two dice, and list all possible
probabilities
for two dice sums.
 How many ways can 4 different
heads
be combined
with 4 different bodies?
 How many ways can 9 fish be
placed
in 2 bowls?
[Not in NYC version]
 How many ways can 5 people shake
hands with
each other, with each person shaking every other hand just once?
 Sample student opinion and
describe
vote data
for 3 possible choices of color for a school uniform.
 Collect, organize, and compare
data
for the
average of 3 shots for each of 9 catapult settings.
 Middle School Statistics and
Probability
Tasks
 Analyze "distance from the pin"
data
for 3
different golfers. Find the lower extreme, lower quartile,
median,
upper quartile, and upper extreme for 10 shots by each golfer.
Develop
a box plot for each golfer's data.
 Find the probability that a
player
can first
predict a coin toss and then roll an 11 or 12, using a pair of
fair
dice. If a player pays $1 and a winner gets $10, find the
"expected"
profit after 24 players.
 High School Statistics
and
Probability
Tasks
 If a zero (0) replaces the four
(4)
on one
die and also replaces the one (1) on a second die, find the probability
of rolling an 8 with this pair of dice. Also determine which
sum(s)
are most likely with this pair of dice. [NYC only]
Middle school task #2 is as difficult as
it
gets for NCEE probability. The NCEE complimented the
student for "realizing
that
his ^{1}/_{24} probability implied that he could
expect
one winner per 24 games played." The student said "out of every 24
players,
one would win." It
appears
that both the student and the NCEE authors believe that one win is sure
to happen every 24 plays.
As for NCEE statistics, there's no
example
involving the arithmetical mean, and there's no mention of standard
deviation
or standard distributions (normal, binomial, and
exponential).
For a comprehensive list of the missing topics in statistics,
data
analysis, and probability, please see the California Mathematics
Content
Standards, Chapter 2 of The
California Mathematics Framework. In particular, see the the
statistics, data analysis, and probability sections for kindergarten
through
grade 7, and see the high school sections for probability and
statistics.
Doing
Math Without Knowing Math Content
Constructivist math educational methods
don't produce students who possess a solid knowledge base of remembered
math content. There are several reasons:
 The shallow content described
above.
 The rejection of the need to
remember specific
math content.
 Constructivists claim that
remembering specific
content is a mistake because such knowledge changes too
rapidly.
They say if you need facts, just look them up. But "doing math"
is
a process that occurs in the brain. Remembered math
knowledge
is pulled together, connected in the mind, and applied to solve
the
current problem. External knowledge "look ups" may be involved,
but
remembered knowledge provides the necessary orienting framework of
background
information. Remembered knowledge leads to the recognition of
what
should be looked up, and it leads to the recognition of how the
lookedup
knowledge can used.
 The rejection of memorization
and
practice.
 Memorization is a learning
device
for loading
information into the brain. This sets up the possibility for
connecting
to other knowledge so that understanding can occur.
 Practice improves knowledge
retention.
It also allows knowledge fragments to be integrated to form
concepts.
Practice is associated with a gradual transition from conscious thought
to automatic use. This frees the conscious mind to focus on other
ideas.
 The rejection of the precise
language of
mathematics.
 The precise language and symbols
of
math provide
a powerful universal vehicle for clear and concise communication.
But the NCEE promotes "natural language" and "personal
language."
They are following the lead of their cofounder Lauren Resnick.
She
recommends the "greater use of
ordinary language, rather than the specialized language and notation of
mathematics in mathematics classroom."
See: Mathematics
as an IllStructured Discipline  Resnick
 The timeconsuming busywork
associated
with NCEE math.
 There's no time for multiple
experiences with
ideas.
 The NCEE attempts to get around
this
by revisiting
the same ideas (such as cubes), but this further limits the content
covered
in NCEE math.
 The emphasis on small group
learning.
 Someone in the group may acquire
the
knowledge.
That's considered sufficient.
 The avoidance of critical
evaluation.
 Although samples of work
from
multiple
students are regularly offered in NCEE math, there are no notes that
might
be considered "grading" a student's performance. Each student
must
be seen as offering a unique, personal way of doing math, and that
student's
efforts must be respected and considered as valuable as any other
student's
work.
With no base of remembered math content,
the
NCEE must redefine the meaning of conceptual understanding, problem
solving
and reasoning. That's the subject of the remainder of this
chapter.
How
the NCEE Redefines Conceptual Understanding in Mathematics
The NCEE's redefines conceptual
understanding
to mean give a concrete demonstration, draw a picture, relate to the
real
world, explain in your own words, say it another way, or do it multiple
ways.
Consider these NCEE "standardsetting"
methods for calculating 62 x 85:
 Method 1: The student
"added
85, 62 times."
 Method 2: "The
student
showed
how the problem could be solved by making 62 circles with 85 stars in
each
circle." The student drew one circle containing 62 stars and
wrote
"I need to make 62 circles like this one to get the problem 62 x 85."
 Method 3: "The
student
drew a
base 10 block array."
 Method 4: The student
wrote:
"First
I multiplied 5 x 62. Next I multiplied 80 x 62." The
partial
products and final sum (310, 4,960, and 5,270) are correctly presented
in standard vertical format. There are no further student
comments.
 Method 5: The student
wrote:
"I broke the problem into 4, got 4 problems, the answers, and added
them
up." The student also wrote (60 x 80) = 4800, (60 x 5) = 300, (2 x 80)
= 160, (2 x 5) = 10, and the total 5,270. There are no further
student
comments.
 Method 6: The student
wrote:
"First I rounded 85 to 90. Then I multiply (sic) 90 x 62. Next I
multiplied
62 x 5. Then I subtracted 310 from 5,580 and got 5,270." The
three
calculations are presented in standard vertical format. There are no
further
student comments.
Methods 1 though 3 exemplify the busywork
that characterizes all levels of NCEE math. Method 1 also
illustrates
a "say it another way" demonstration of conceptual
understanding.
Constructivist math educators are pleased when students describe
multiplication
as continued addition and division as continued
subtraction.
Since they never get beyond whole numbers, they're happy with this
conceptualization
at the high school level. But this characterization isn't even
ideal
for whole numbers, and it's certainly wrong for fractions.
Note: Amazingly, the
NCTM
attempts to explain the division of fractions as continued subtraction
in their new version of the NCTM Standards.
Eventually children need to recognize the
limitations of whole numbers. They're not closed under
subtraction,
and they're not closed under division. Eventually children need
to
understand how these defects are remedied by expanding to the integers
and by expanding to the rational numbers. They need to understand
why operations with rational numbers must be defined as they are in
order
preserve fundamental number properties when addition and multiplication
are extended from the integers to the rational numbers.
Eventually
they need to understand addition and subtraction as inverse operations
relative to the additive identify element, the number 0, and they need
to understand multiplication and division as the inverse operations
relative
to the multiplicative identity element, the number 1. Then
they can leave behind the incomplete world of whole numbers, where some
subtractions can't be performed, and some divisions leave troubling
"remainders."
Now let's consider Methods 4 through 6
for calculating 62 x 85. They're all justified by ideas that
underlie
the standard algorithm for multidigit multiplication. Consider
Method
4. All of the following key ideas are involved:
 Decomposing a number relative to
place
value
 Example: 85 is expanded as
80
+ 5.
[When we express 85 as (80 + 5) or as (8 x 10) + 5, we are expanding 85
using its place value definition.]
 Substitution of equals for equals
 Example: 62 x 85 = 62 x
(80 +
5)
 The distributive law
 Example: 62 x (80 + 5) =
(62 X
80) +
(62 x 5)
 The commutative law of
multiplication
 Regrouping relative to place value
representation
 Example: Recognizing that
5 x
62 = 310
requires "carrying a 1".
Students demonstrate genuine conceptual
understanding
when they can supply the mathematical reason for each step in a
method.
But NCEE students don't do that. For Methods 4 through 6,
we
are briefly told what the student did, but never why.
There's no evidence that these students know why. The five major
ideas just listed are never explicitly mentioned in NCEE math.
Place value number representation and
the
distributive law are the two most powerful concepts in elementary
mathematics.
Both are suppressed in NCEE math. The NCEE always uses the
blanket
statement "broke the number apart" to describe both applications of the
distributive law and decompositions relative to place value. They
avoid mentioning carrying and borrowing by choosing numbers so that
neither
is required, or by transforming the problem into an equivalent
problem
which doesn't require either, or by providing the answer with no
explanation
of the carrying or borrowing details. This last technique was
used
for Methods 4 through 6.
Although Methods 4 through 6 rely on
ideas
that justify the standard algorithm for multidigit multiplication,
standard
computational methods are never demonstrated in NCEE math.
Constructivist
math educators reject them, claiming that students follow the steps of
standard procedures in a "parrotlike" fashion, not really
understanding
what they are doing. There's a hint of truth in this
complaint
because standard procedures hide their own justifying reasons.
Students may forget why they work. Occasionally, good
teachers push students out of automatic computation mode to
remind students of the the underlying place value logicl.
To explain the logic behind
the standard procedure for calculating 62 x 85, teachers usually remind
students that the 6 x 5 step is really 60 x 5, etc. But
eventually, as students gain mathematical maturity, they need the full
explanation. First
62 and 85 are expanded using their place value definitions. Then
the distributive, commutative, and associative laws are used to show:
 62 x 85 = ((6 x 10) + 2))
x
((8 x 10)
+ 5))

=((6 x 10) + 2)) x (8 x 10) + ((6 x 10) + 2)) x 5

= (6 x 10) x (8 x 10) + 2 x (8 x 10) + (6 x 10) x 5 + (2 x 5)

= (6 x 8) x 100 + (2 x 8) x 10 + (6 x 5) x 10 + (2
x 5) = (4 x 1000) + ( 8 x 100) + (1 x 100) + (6 x 10) + (3 x 100) + (1 x 10)

= (4 x 1000) + (12
x 100) + (7 x 10) = (4 x 1000) + (1 x 1000) + (2 x 100) + (7
x 10)

= (5 x 1000) + (2 x 100) + (7 x 10) [Expanded place value form]

= 5270 [Standard (compressed) place value form]
The bold type points to the four
singledigit
multiplication facts in line 4. The standard algorithm's compact
vertical
format hides these details. It exploits the power of place
value representation to allow efficient calculations in all
cases.
Methods 4, 5, and 6 may appear attractive for the product of two
2digit
whole numbers, but they become unwieldy for more complex
calculations.
Consider Method 5. The number of partial products for this method
equals the product of the number of digits in the two factors. Thus,
for
a 3digit times 4digit multiplication, there will be 12 partial
products,
versus only three for the standard algorithm.
Note the small amount of knowledge that
must be remembered in order to carry out the standard algorithm
multiplication
of any pair of multidigit numbers: just singledigit multiplication
facts,
singledigit addition facts, and knowing how to
carry.
More importantly, since the standard algorithm involves repeating a
sequence
of simple steps, it can, with sufficient practice, be carried out
automatically.
This frees the conscious mind for more advanced learning. But
constructivist
math educators aren't interested. They point to calculators for
anything
beyond small whole numbers. They don't admit that mastery
of
algebra isn't possible without prior mastery of the standard methods of
genuine pencilandpaper arithmetic.
There's another important fact about
conceptual
understanding: it deepens as newly acquired knowledge provides an
increasingly richer frame of reference. Consider two levels of
understanding
of 2 + 2 = 4. Students first understand this fact concretely,
perhaps
using plastic chips. This is fine for the first grade. By
the
end of the 5th grade, the student should know enough to understand the
following proof:
 4 = 3 +
1
 Definition of 4
 3 = 2 +
1
 Definition of 3
 4 = (2 +1) + 1

Substitution
of equals for equals
 4 = 2 + (1 + 1) 
Associative
law of addition
 4 = 2 +
2
 Definition of 2 and substitution of equals for equals
 2 + 2 =
4
 Symmetric property of equality
Good teachers balance two major goals:
moving
on to higher levels of complexity, while working to maximize conceptual
understanding to the greatest extent possible at the current grade
level.
They know that important ideas must sometimes be pushed into the
background.
They work to regularly bring them back into the foreground to allow for
deeper understanding relative to the increasingly larger context of
knowledge
acquired by the student (stored in the brain), while migrating up
the math learning curve.
The NCEE is satisfied with entry level
math. Their "vision" is limited to the needs of everyday
life.
The concept of prerequisite knowledge is never mentioned. They're
not concerned with setting the stage for learning more advanced
math.
Many of their high school math examples belong at the elementary school
level. They claim to emphasize conceptual understanding, but give
no evidence that they understand how math ideas are connected.
They
appear blind to the verticallystructured nature of the math knowledge
domain.
How
the NCEE Redefines Math Problem Solving and Math Reasoning
The NCEE claims that 24 WS&C sets
demonstrate
problem solving and reasoning skills. But this isn't about
applying
remembered math knowledge or deductive reasoning. The NCEE
problem solving skills aren't specific to math. Without
remembered
content knowledge, the NCEE must emphasize contentindependent methods
such as making a list, developing a chart or table, drawing a picture
or
diagram, and trial and error (or guess and check).
Consider these counting tasks listed in
the first section:
 How many ways can 4 different
heads
can be
combined with 4 different bodies?
 How many ways can 9 fish can be
placed
in
2 bowls? [not in NYC version]
 How many ways can 5 people can
shake
everyone
else's hand just once?
 "How many segments are needed
to
connect
5 points? 6 points? 8 points? 10 points? 30
points?
100 points? n points?"
NCEE math is persistently handson. The
same
low level methods are demonstrated for all four problems. The
student
draws, lists, counts, and writes lengthy explanations. The
NCEE does hint at more efficient counting strategies. For task 1
they say that the student invented a "multiplicative formula" to
justify the 16 drawing solution. The student earned this
lofty
recognition by writing "4 monster heads x 4
different
bodys (sic) to put them on" equals "16 different ways." But
the NCEE isn't setting the stage to introduce the Fundamental Counting
Principle (FCP). Although the FCP efficiently yields the
answer
for all these counting tasks, this basic unifying idea is never
mentioned
at any level in NCEE math.
The NCEE avoids recognizing unifying
ideas
and general methods. Consider how they have suppressed the ideas
that connect tasks 3 and 4. Tasks 3 is mathematically identical
to
the first part of task 4. But the NCEE never encourages students
to ask "is this problem similar to one that I've previously
solved?"
More generally, there's no interest in recognizing general methods that
can be used to solve an entire class of problems. It's not
that the NCEE doesn't know that such methods exist. For task 4
the
students sketched and counted for several cases and said "then I found
this ((n ÷ 2)  .5) x n as the formula and it
worked."
The NCEE commented that "the
student's
phrase, 'then I found this,' leaves the reader to wonder, 'How'." There's
no further NCEE comment and no recognition that the student
"discovered"
the general formula for the number of combinations of n objects taken 2
at a time.
Advanced counting methods, such as
permutations
and combinations, aren't covered anywhere in NCEE math. Why
not?
Perhaps because these topics require knowledge about fractions.
But
the "wonder how" comment may be revealing. Even the NCEE may
recognize
the stretch necessary to sell the idea that the student discovered the
formula for the number of combinations of n objects taken 2 at a
time.
They must recognize that no one will buy student discovery of more
general
counting formulas. Their constructivist faith forces them to
avoid
concepts that can't be sold as "discovered" via pattern
recognition.
They stick to their comfort zone: entry level problems that can be
solved
with contentindependent skills.
What about NCEE math reasoning?
You
won't find deductive proofs in NCEE math, and you won't anything like
the
explanation given above for the ideas behind the standard algorithm for
multidigit computation. The NCEE constructivist reasoning
methods
are limited to handson demonstrations, pattern recognition, and a weak
form of inductive reasoning (generalizing from a small number of
examples).
There's much busywork with manipulatives, lists, charts, and supplies
used
to paint, draw, cut, tape, and paste.
Consider the NCEE method for testing
Eddies'
claim that the surface area of a cube is always twice the volume of the
cube.
The student already knew the
claim
was true for the case of edge length 3. Multiple other edge
lengths
were tried, with drawings for each case. The student then said
that
Eddie's claim wasn't true "because other examples don't follow that
rule."
An NCEE note says "several
examples
are given to counter Eddie's claim, not just one example. Not
only
is Eddie's claim not true for all cubes, it is false for most cubes."
There's no further NCEE comment, but we have a few:
 Once the student determines that
the
claim
isn't generally true, the focus should shift to finding a compact
characterization
of the cases where it is true. That's easily done here by
considering
the algebraic formulation of the conjecture. If x denotes edge
length,
it's equivalent to saying that 2x^{3 }= 6x^{2}.^{ }
But this is true if and only if x= 3 or x = 0. Neither the
student
nor the NCEE noted this fact.
 Once the student determines that
the
claim
isn't generally true, the focus should shift to finding other patterns
involving volume and surface area. But no other patterns
are
mentioned
 No one noticed that Eddie said
"always," or
perhaps they don't know that one counterexample is sufficient.
 If NCEE students had found
Eddie's
claim to
be true for all other cases tested, one suspects that they would have
concluded
that it was always true. It appears that four examples
suffices
as a proof in NCEE math.
The praise for multiple counterexamples
isn't
the only illustration of NCEE fuzzy logic and imprecise language.
Consider the task involving "linear" and "nonlinear"
patterns.
According to the NCEE, "linear" is defined as "repeating," and
"nonlinear"
is defined as "grow or lower itself." It appears that nonlinear
doesn't mean not linear! Of course there's no problem if
"not
repeating" means "grow or lower itself." There's really no way to
know, since the NCEE hasn't defined either "repeating" or "grow or
lower
itself."
Next?
Please
click on one of the following links.
Chapter 2: How
the NCEE Limits Elementary School Math
Chapter 3: How
the NCEE Limits Middle School Math
Chapter 4: How
the NCEE Limits High School Math
Copyright
20022011 William G. Quirk, Ph.D.