Understanding the Revised NCTM Standards

Arithmetic is Still Missing!

by Bill Quirk  ( wgquirk@wgquirk.com)

Contrary to Recent Reports, the NCTM Has Not Changed Its Philosophy

On April 12, 2000, The National Council of Teachers of Mathematics (NCTM)  released Principles and Standards for School Mathematics (PSSM), a 402 page revision of  the NCTM Standards. The next day The New York Times reported: "In an important about-face, the nation's most influential group of mathematics teachers announced yesterday that it was recommending, in essence, that arithmetic be put back into mathematics, urging teachers to emphasize the fundamentals of computation rather than focus on concepts and reasoning."  It was further reported that "the council added strong language to its groundbreaking 1989 standards, emphasizing accuracy, efficiency and basic skills like memorizing the multiplication tables."

Compare the preceding New York Times quotes to the following contradictory quote, published by the NCTM (in the third PDF file, Commonsense Facts to Clear the Air) under "News and Hot Topics" at  NCTM Speaks Out: Setting the Record Straight about changes in Mathematics Education.

When calculators can do multidigit long division in a microsecond, graph complicated functions at the push of a button, and instantaneously calculate derivatives and integrals, serious questions arise about what is important in the mathematics curriculum and what it means to learn mathematics. More than ever, mathematics must include the mastery of concepts instead of mere memorization and the following of procedures. More than ever, school mathematics must include an understanding of how to use technology to arrive meaningfully at solutions to problems instead of endless attention to increasingly outdated computational tedium.   -NCTM,  Commonsense Facts to Clear the Air
It's clear that The New York Times was fed misleading NCTM propaganda, perhaps designed to placate "math wars" opponents.  Not surprisingly, we will show here that the NCTM has not rediscovered arithmetic.  Similar to the original NCTM Standards, PSSM is vague about the major components of arithmetic mastery:
    1. Memorization of of basic number facts
    2. Mastery of the standard algorithms of multidigit computation.
    3. Mastery of fractions
The NCTM has toned down the constructivist language, but they still stress content-independent "process skills" and student-centered "discovery learning".  Similar to the NCTM Standards, PSSM emphasizes manipulatives, calculator skills, student-invented methods, and simple-case methods.

Although PSSM contains five "Connections" sections,  there continues to be no acknowledgement of the vertically-structured nature of  mathematics.  Mastery of math requires a step-by-step build up (in the brain) of specific content knowledge.  PSSM omits this aspect of the "connections" within mathematics.  The idea conveyed by the following example is not found in PSSM.

Example: Migrating up the math learning curve

Each of the following skills serves as a preskill for acquiring all higher skills. To move up to the next skill level, the student must remember all preskills.

  1. The ability to instantly recall basic multiplication facts
  2. The ability to factor integers
  3. The ability to reduce a fraction to lowest terms.
The NCTM says they want to maximize "understanding", but they still fail to recognize that specific math content must first be stored in the brain as a necessary precondition for understanding to occur.  Although rarely the preferred method, intentional memorization is sometimes the most efficient approach.  The first objective is to get it into the brain!  Then newly remembered math knowledge can be connected to previously remembered math knowledge and understanding becomes possible. You have to "know math" before you can "understand math", "do math", or "solve math problems."

We conclude this introductory section by noting that there is evidence of a battle within the NCTM, with some voices crying out for genuine arithmetic. These voices were heard in the Principles and Standards for School Mathematics: Discussion Draft (PSSM Draft), published in October, 1998. At later points in this document you will find quotes from both PSSM and PSSM Draft. The quotes from PSSM Draft do not appear in PSSM, the final version published in April, 2000.  The voices of reason have been largely silenced!  Here is an example of the silencing.

Technology: PSSM Draft vs. PSSM

However, access to calculators does not replace the need for students to learn and become fluent with basic arithmetic facts, to develop efficient and accurate ways to solve multidigit arithmetic problems, and to perform algebraic manipulations such as solving linear equations and simplifying expressions.  -PSSM Draft, Page 43

Technology should not be used as a replacement for basic understandings and intuitions; rather it should be used to foster those understandings and intuitions.  -PSSM, page 24

They Still Call It "Arithmetic"

Mastery of Basic Facts or Derive Them When Needed?

Similar to the original NCTM Standards, PSSM fails to clearly acknowledge that the ability to instantly recall basic number facts is an essential preskill, necessary to free up the mind, first for mastery of the standard algorithms of multidigit computation, and next for mastery of fractions. Then, once this knowledge is also instantly available in memory, the mind is again free to focus on the next level, algebra.

As the quotes below show, the PSSM Draft emphasized quick recall, but it appears that PSSM has reverted to the NCTM Standards idea that "the ability to efficiently derive" is preferable to "the ability to instantly recall."

Quotes From The PSSM Draft:  Recall Number Facts Quickly

Most students should be able to recall addition and subtraction facts quickly by the end of grade 2 and recall multiplication and division facts with ease and facility by the end of grade 4.  -PSSM Draft, Page 51

A certain amount of practice is necessary to develop fluency with both basic fact recall and computation strategies for multi-digit numbers.  Anderson, Reder, and Simon (1996) point out that practice is clearly essential for acquiring cognitive skills of almost any kind. -PSSM Draft, Page 114

Quotes From PSSM:  Derive Number Facts Quickly

Fluency with basic addition and subtraction number combinations is a goal for pre-K-2 years.  By fluency we mean that students are able to compute efficiently and accurately with single-digit numbers.   -PSSM, Page 84

When students leave grade 5, they should be able to . . . efficiently recall or derive the basic number combinations for each operation.  -PSSM, Page 149

Fluency with the basic number combinations develops from well understood meaning for the four operations and from a focus on thinking strategies. -PSSM, Page 152-153

If by the end of the fourth grade, students are not able to use multiplication and division strategies efficiently, then they must either develop strategies so that they are fluent with these combinations or memorize the remaining "harder" combinations. -PSSM, Page 153

The changes may be subtle, but notice that fluency with basic number facts is defined as the ability to "compute efficiently and accurately."   It does not mean the ability to instantly recall.  Also, in PSSM it's "recall or derive" by the end of grade 5, not "recall" by the end of grade 4 as recommended in the PSSM Draft.  Basic number fact "thinking strategies" appear to be preferred by the writers of PSSM.  They give grudging admission that memorization may be necessary.  There is some discussion of activities to teach fact relationships, but there is no discussion of mastery activities to facilitate fact memorization.

Mastery of Standard Algorithms or Student-Invented Algorithms?

Considering that the NCTM appears to prefer basic number fact derivation strategies, it's not surprising that they also appear to prefer student-invented algorithms for multidigit computation. Here are some  relevant quotes from PSSM: This last quote contains a hint of truth.  If a student is given a problem and allowed to struggle for a while, trying to solve the problem, then the student becomes motivated to listen and learn about the most efficient, general solution to the problem.  This is the essence of the lesson method used in Japan.  The mistake is to elevate the value of "easier to understand" student-invented methods, while not stressing the power, mathematical importance, and universal acceptance of the efficient, general "algorithms traditionally taught in school."

Efficient, Accurate, and (Possibly) General Methods

On page 32 of PSSM,  the term "computational fluency" is defined as "having and using efficient and accurate methods for computing".  Later on the same page, we are told that students should "see the usefulness of methods that are efficient, accurate, and general."  On page 87  we are told  "Teachers also must decide what new tasks will challenge students and encourage them to construct strategies that are efficient and accurate and that can be generalized."

Has the definition of computational fluency been (appropriately) expanded to include "general"?   No, the original definition of computational fluency, including only efficient and accurate,  is restated on pages 79 and 153.   It appears that some PSSM writers recognized that all three characteristics contribute to the power of the standard algorithms of arithmetic.  But the standard algorithms are not mentioned in this context (page 87).  Instead, on this page we are advised:  "As students encounter problem situations in which computations are more cumbersome or tedious, they should be encouraged to use calculators to aid in problem solving."

The Truth is in The Examples

Within the five "Number and Operations" sections, PSSM includes (only) three illustrations of  multidigit computation. All are "student-invented" strategies.  On page 85 we learn "In some cases, their strategies for computing will be close to conventional algorithms; in other cases, they will be quite different."  There is no discussion of the accuracy, efficiency or generality of any method found in these three illustrations. Apparently those who wrote about "computational fluency" failed to communicate with those who developed the examples.
  1. Figure 4.3 on page 85 presents six student solutions for computing 25 + 37. Student 2's method utilized 12 "tallies" (four vertical marks crossed by a horizontal mark) followed by two additional vertical marks, with the 12 tallies identified by  5 written above the first tally, 10 written above the second tally, 15 written above the third tally, and so on until 62 is written above the concluding pair of vertical lines.  The NCTM is pleased with the "completeness" of Student 2's thinking.  Student 4 correctly used the standard algorithm for addition, but the NCTM appears not to notice, even remarking that student 4's thinking is "not as apparent."
  2. Figures 4.4 and 4.5 on page 86 describe student strategies for computing 153 + 273. Randy's method is described first.  He used beans, bean sticks (10 beans), and rafts of bean sticks (100 beans).  The "conventional algorithm" is used successfully by some nameless students,  but unsuccessfully by other nameless students.  "Becky finds the answer using mental computation and writes nothing down except her answer."    Subtle, but effective.  Randy and Becky are worth recognizing by name.
  3. Page 153 presents two student solutions for dividing 728 by 34.  Henry used the method of repeated subtraction of multiples of 10, which he apparently invented. Michaela used long division, which she apparently invented. Mrs. Sparks "saw the relationship between the two methods described by the students, but she doubted that any of her students would initially see these relationships".  This is a surprising lack of confidence, considering the remarkable discovery abilities demonstrated by Henry and Michaela.
    1. (We used "long division" to briefly describe the method used by Michaela, but the phrase long division in not found in PSSM.)
In this last illustration, we again we have a hint of the lesson method used in Japan.  But we learn that Mrs. Sparks objective is to "help the students understand, explain, and justify their computational strategies,"  rather than working to achieve closure by  connecting the two methods and teaching long division, emphasizing the efficiency and generality of the long division algorithm.

Mastery of Fractions or Simple-Case Methods for "Familiar Fractions"?

Consistent with the lukewarm treatment of number fact recall, PSSM fails to emphasize the importance of the ability to factor integers, and PSSM never discusses any of the details related to the addition, subtraction, multiplication, division, and simplification of fractions.  The phrase "common denominator" is not found in PSSM.

The NCTM says they want students to "develop and analyze algorithms for computing with fractions" and "develop and use strategies to estimate the results of rational-number computations and judge the reasonableness of the results."  - PSSM, Page 214.

PSSM's treatment of fractions offers just two illustrations, one for comparing fractions,  and the other for dividing fractions.  Both are simple-case methods, and neither is efficient or general.

PSSM on "invert and multiply""

The division of fractions has traditionally been quite vexing for students.  Although "invert and multiply" has been a staple of conventional mathematics instruction and although it seems to be a simple way to remember how to divide fractions, students have for a long time had difficulty doing so.  Some students forgot which number is to be inverted, and others are confused about when it is appropriate to apply the procedure.  A common way of formally justifying the "invert and multiply" procedure is to use sophisticated arguments involving the manipulation of algebraic rational expressions—arguments beyond the reach of many middle-grade students.  This process can seem very remote and mysterious to many students. Lacking an understanding of the underlying rationale, many students are therefore unable to repair their errors and clear up their confusions about division of fractions on their own. An alternate approach involves helping students . . . understand the meaning of division as repeated subtraction."  -PSSM, page 219   (underline added)

For the sophisticated arguments, see pages 2-3 in Basic Skills Versus Conceptual Understanding: A Bogus Dichotomy in Mathematics Education, By H.Wu

PSSM recommends that probability be covered in every grade, offering five PSSM sections on "Data Analysis and Probability".  The NCTM is not bothered by the fact that any meaningful discussion of elementary probability requires prior mastery of fractions.

Liping Ma's Book: U.S. Elementary School Teachers Don't Understand Arithmetic

U.S. elementary school teachers frequently don't understand the underlying "whys" of arithmetic, but the same can't be said of Chinese math teachers.  This is one message of the new book,  Knowing and Teaching Elementary Mathematics (KTEM) by Liping Ma.  

U.S. teachers fared poorly when asked questions related to the teaching of:

  1. Subtraction with regrouping
  2. Multidigit multiplication
  3. Dividing fractions
Don't blame the teachers!  None of these topics appear in PSSM or the PSSM Draft.  The term "subtraction with regrouping (or renaming)" is never used.  There is one example of multidigit multiplication, found on page 220 of PSSM. The text states: "the cost of 1.37 pounds of cheese at $2.95 a pound might be estimated, although a calculator would probably be the preferred tool."  The failure to cover division by fractions was discussed above.

Liping Ma's book is referenced in two ways in PSSM.  In each case her ideas have been misused:

  1. Ma uses the phrase "profound understanding of fundamental mathematics" (PUFM).  A teacher who possesses PUFM has a comprehensive understanding of the "network of procedural and conceptual topics" that comprise elementary mathematics. Such a teacher "is able to reveal and represent connections among mathematical concepts and procedures to students." -Ma, Page 124
  2. The terms "compose" and "decompose" appear frequently in PSSM and in KTEM (but not in the PSSM Draft, which preceded the publication of KTEM). In KTEM these words are only used relative to place value.  If "compose" appears alone in KTEM, it's always shorthand for "compose a unit of higher value" (old term is carrying).  If "decompose" appears alone in KTEM, it's always shorthand for "decompose a unit of higher value" (old term is borrowing).  PSSM never uses these terms this way. PSSM never discusses carrying or borrowing  (or any other equivalent terms).
The NCTM says in boldface  "The status quo of traditional mathematics isn't working." Liping Ma's book shows that the real problem is the failure to correctly teach "traditional mathematics."

Not Subtle in Connecticut:  Arithmetic is Obsolete

It's difficult to see what's missing, but in Connecticut it's in boldface.  The Connecticut Mastery Test (CMT), Third Generation, Mathematics Handbook states:  Connecticut teachers are told they have:
Permission to Omit. An amazing amount of time and energy is still expended by you and by your students on increasingly obsolete skills.  Teachers need to give each other permission to skip textbook pages that no longer serve a useful purpose.  So give yourself and your colleagues permission to omit such things as:

Role of Mathematicians: Advice solicited,  Advice Received, Advice Ignored

The NCTM solicited advice from mathematicians:
In order to provide for this complex advisory function, the NCTM petitioned each of the professional organizations of the Conference Board of the Mathematical Sciences (CBMS) to form an Association Review Group (ARG) that would respond, in stages, to a series of substantial and focused questions framed by the Principles and Standards writing group in the course of its work.   -PSSM, Page xv
The NCTM received excellent input (see examples below), but ignored it. None of the several response reports (including the two quoted below) are referenced in either PSSM or the PSSM Draft.

One set of questions, via a letter from Joan Ferrini-Mundy and Mary Lindquist on April 1, 1997, asked:

  1. What is meant by "algorithmic thinking"?
  2. How should the Standards address the nature of algorithms in their more general mathematical context?
  3. How should the Standards address the matter of invented and standard algorithms for arithmetic computation?
  4. What is it about the nature of algorithms that might be important for children to learn?
Roger Howe, Professor of Mathematics at Yale University, responded in the American Mathematical Society NCTM2000 Association Resource Group Second Report in June, 1997 (See following first report via pdf file at Reports of AMS ARG ). Here are five excerpts from Professor Howe's response:
An important feature of algorithms is that they are automatic and do not require thought once mastered. Thus learning algorithms frees up the brain to struggle with higher level tasks.  On the other hand, algorithms frequently embody significant ideas, and understanding of these ideas is a source of mathematical power.  -Howe, Page 273

 . . . we suspect it is impractical to ask all children personally to devise an accurate, efficient, and general method for dealing with addition of any numbers—even more so with the other operations. Therefore, we hope that experimental periods during which private algorithms may be developed would be brought to closure with the presentation of and practice with standard algorithms. Also we hope care would be taken to ensure that time spent developing and testing private algorithms will not significantly slow overall progress. -Howe, Page 274

Standard algorithms may be viewed analogously to spelling:  to some degree they constitute a convention, and it is not essential that students operate with them from day one or even in their private thinking; but eventually, as a matter of mutual communication and understanding, it is highly desirable that everyone (that is nearly everyone—we recognize that there are always exceptional cases) learn a standard way of doing the four basic arithmetic operations. -Howe, Page 275

We do not think it wise for students to be left with untested private algorithms for arithmetic operations—such algorithms may only be valid for some subclass of problems.  The virtue of standard algorithms—that they are guaranteed to work for all problems of the type they deal with—deserves emphasis.  -Howe, Page 275   (bold added)

We would like to emphasize that the standard algorithms of arithmetic are more than just "ways to get the right answer"—that is, they have theoretical as well as practical significance.  For one thing, all the algorithms of arithmetic are preparatory for algebra, since there are (again, not be accident, but by virtue of construction of the decimal system) strong analogies between arithmetic of ordinary numbers and arithmetic of polynomials. The division algorithms is also significant for later understanding of real numbers. -Howe, Page 275
 

Kenneth Ross, Professor of Mathematics, University of Oregon, also responded to these four questions in The Mathematical Association of America's Second Report from the Task Force.  Here are three excerpts from Professor Ross's June 17, 1997 report:
The NCTM Standards emphasize that children should be encouraged to create their own algorithms, since more learning results from "doing" rather than "listening" and children will "own" the material if they create it themselves. We feel that this point of view has been over-emphasized in reaction to "mindless drills."  It should be pointed out that in other activities in which many children are willing to work hard and excel, such as sports and music, they do not need to create their own sports rules or write their own music in order to "own" the material or to learn it well. In all these areas, it is essential for there to be a common language and understanding. Standard mathematical definitions and algorithms serve as a vehicle of human communication. In constructivistic terms, individuals may well understand and visualize the concepts in their own private ways, but we all still have to learn to communicate our thoughts in a commonly acceptable language. -Ross, Page 1

The starting point for the development of children's creativity and skills should be established concepts and algorithms. As part of the natural encouragement of exploration and curiosity, children should certainly be allowed to investigate alternative approaches to the task of an algorithm. However, such investigation should be viewed as motivating, enriching, and supplementing standard approaches. Success in mathematics needs to be grounded in well-learned algorithms as well as understanding of the concepts. None of us advocates "mindless drills."  But drills of important algorithms that enable students to master a topic, while at the same time learning the mathematical reasoning behind them, can be used to great advantage by a knowledgeable teacher. Creative exercises that probe students' understanding are difficult to develop but are essential.  -Ross, Page 1-2

The challenge, as always, is balance. "Mindless algorithms" are powerful tools that allow us to operate at a higher level. The genius of algebra and calculus is that they allow us to perform complex calculations in a mechanical way without having to do much thinking. One of the most important roles of a mathematics teacher is to help students develop the flexibility to move back and forth between the abstract and the mechanical. Students need to realize that, even though part of what they are doing is mechanical, much of mathematics is challenging and requires reasoning and thought. -Ross, Page 2

Published in American Educator (AFT), But Also Ignored by the NCTM

Below you will find links to two articles that were published in the Fall 1999 issue of American Educator/American Federation of Teachers.  Both of these were available well prior to the release of PSSM.  Neither is referenced in PSSM.

Basic Skills Versus Conceptual Understanding: A Bogus Dichotomy in Mathematics Education, By H.Wu, Professor of Mathematics, University of California, Berkeley

Please see Professor Wu's discussion of the division of fractions and the standard algorithms.  The following two quotes are from the beginning of Professor Wu's article.

Education seems to be plagued by false dichotomies. Until recently, when research and common sense gained the upper hand, the debate over how to teach beginning reading was characterized by many as "phonics vs. meaning."  It turns out that, rather than a dichotomy, there is an inseparable connection between decoding—what one might call the skills part of reading—and comprehension. Fluent decoding, which for most children is best ensured by the direct and systematic teaching of phonics and lots of practice reading, is an indispensable condition of comprehension. -Wu, Page 1

"Facts vs. higher order thinking" is another example of a false choice that we often encounter these days, as if thinking of any sort—high or low—could exist outside of content knowledge. In mathematics education, this debate takes the form of “basic skills or conceptual understanding.” This bogus dichotomy would seem to arise from a common misconception of mathematics held by a segment of the public and the education community: that the demand for precision and fluency in the execution of basic skills in school mathematics runs counter to the acquisition of conceptual understanding. The truth is that in mathematics, skills and understanding are completely intertwined. In most cases, the precision and fluency in the execution of the skills are the requisite vehicles to convey the conceptual understanding. There is not 'conceptual understanding' and 'problem-solving skill' on the one hand and 'basic skills' on the other. Nor can one acquire the former without the latter. - Wu, Page 1

 Knowing And Teaching Elementary Mathematics, By Richard Askey, John Boscom Professor of Mathematics, University of Wisconsin-Madison

The title of this article is also the title of the new book by Liping Ma. Please see Professor Askey's discussion of the division of fractions.  The following quote is from Professor Askey's article.

Copyright 2000-2011 William G. Quirk, Ph.D.