A Summary View of NCEE Math

Chapter 1 of How the NCEE Redefines K-12 Math

Includes the New York City Modifications

by Bill Quirk (E-Mail: wgquirk@wgquirk.com)

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 3-volume 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 state-wide 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 K-12 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 K-12 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 content-independent 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 K-5 computational tasks:
  1. Share 25 objects as "equally as possible" among four friends.
  2. Calculate 63 x 46.
  3. Calculate 522 - 367 in two different ways.
  4. Calculate 87 x 9 in two different ways.
  5. Demonstrate multiple ways to calculate 62 x 85.
  6. Share 3 bags of M & M's equally among four children.  Each bag contains 52 M&M's.
  7. Calculate 13 x 14.
There are no NCEE K-12 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:" For the visible small cubes formula (high school task #4),  the student added  n2, n2  - n, and n2 -2n + 1 to get the sum 3n2 - 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".

  1. "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?"
  2. Given 7 equations involving 10 variables,  where each variable represents a unique digit between 0 and 9, find the value of all 10 variables.
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. 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:" 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:

  1. The shallow content described above.
  2. The rejection of the need to remember specific math content.
  3. The rejection of memorization and practice.
  4. The rejection of the precise language of mathematics.
  5. The time-consuming busywork associated with NCEE math.
  6. The emphasis on small group learning.
  7. The avoidance of critical evaluation.
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 "standard-setting" methods for calculating 62 x 85:

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:

  1. Decomposing a number relative to place value
  2. Substitution of equals for equals
  3. The distributive law
  4. The commutative law of multiplication
  5. Regrouping relative to place value representation
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 "parrot-like" 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:

    1. 62 x 85  = ((6 x 10) + 2)) x ((8 x 10) + 5))
    2.                =((6 x 10) + 2)) x (8 x 10) + ((6 x 10) + 2)) x 5
    3.                = (6 x 10) x (8 x 10) + 2 x (8 x 10) + (6 x 10) x 5 + (2 x 5)
    4.                = (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)
    5.                = (4 x 1000) + (12 x 100)  + (7 x 10) = (4 x 1000) + (1 x 1000) + (2 x 100) + (7 x 10)
    6.                = (5 x 1000) + (2 x 100) + (7 x 10)    [Expanded place value form]
    7.                = 5270    [Standard (compressed) place value form]
The bold type points to the four single-digit 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 2-digit 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 3-digit times 4-digit 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 single-digit multiplication facts, single-digit 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 pencil-and-paper 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:

    1. 4 = 3 + 1            -- Definition of 4
    2. 3 = 2 + 1            -- Definition of 3
    3. 4 = (2 +1) + 1    -- Substitution of equals for equals
    4. 4 = 2 + (1 + 1)   -- Associative law of addition
    5. 4 = 2 + 2            -- Definition of 2 and substitution of equals for equals
    6. 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 vertically-structured 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 content-independent 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:

  1. How many ways can 4 different heads can be combined with 4 different bodies?
  2. How many ways can 9 fish can be placed in 2 bowls?  [not in NYC version]
  3. How many ways can 5 people can shake everyone else's hand just once?
  4. "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 hands-on. 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  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 content-independent 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 hands-on 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:
  1. 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 2x3 = 6x2.    But this is true if and only if x= 3 or x = 0.  Neither the student nor the NCEE noted this fact.
  2. 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
  3. No one noticed that Eddie said "always," or perhaps they don't know that one counterexample is sufficient.
  4. 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 "non-linear" patterns.   According to the NCEE,  "linear" is defined as "repeating," and "non-linear" is defined as "grow or lower itself."  It appears that non-linear 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 2002-2011 William G. Quirk, Ph.D.