**Contrary to Recent Reports, the NCTM Has Not Changed Its Philosophy****They Still Call It "Arithmetic"****Mastery of Basic Facts or Derive Them When Needed?****Mastery of Standard Algorithms or Student-Invented Algorithms?****Mastery of Fractions or Simple-Case Methods for "Familiar Fractions"?****Liping Ma's Book: U.S. Elementary School Teachers Don't Understand Arithmetic****Not Subtle in Connecticut: Arithmetic is Obsolete****Role of Mathematicians: Advice Solicited, Advice Received, Advice Ignored****Published in American Educator (AFT), But Also Ignored by The NCTM**

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of how to use technology to arrive meaningfully at solutions to problems instead of endless attention to increasingly outdated computational tedium.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,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:Commonsense Facts to Clear the Air

- Memorization of of basic number facts
- Mastery of the standard algorithms of multidigit computation.
- Mastery of fractions

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.__

__Exampl__e: **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.

- The ability to instantly recall basic multiplication facts
- The ability to factor integers
- The ability to reduce a fraction to lowest terms.

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.

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 43Technology 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

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."

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 51A 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

Fluency with basic addition and subtraction number combinations is a goal for pre-K-2 years. ByThe changes may be subtle, but notice thatfluencywe mean that students are able to compute efficiently and accurately with single-digit numbers. -PSSM, Page 84When 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

- In the past, common school practice has been to present a single algorithm
for each operation. However, more than one efficient and accurate
computational algorithm exists for each arithmetic operation. In
addition, if given the opportunity, students naturally invent methods to
compute that make sense to them. -PSSM, Page 153

Many students are likely to develop and use methods that are not the same
as the conventional algorithms (those widely taught in the United
States). For example, many students and adults use multiplication to
solve division problems or add starting with the largest place rather than
with the smallest. The conventional algorithms for multiplication and
division should be **investigated** in grades 3 - 5 **as one**
efficient** way** to calculate. -PSSM, Page 155 (bold emphasis
added)

Students’ understanding of computation can be enhanced by developing their own methods and sharing them, explaining why their methods work and are reasonable to use, and then comparing their methods with the algorithms traditionally taught in school. In this way, students can appreciate the power and efficiency of the traditional algorithms and also connect them to student-invented methods that may sometimes be less powerful or efficient but are often easier to understand. -PSSM, Page 220

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."

- 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."
- 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.
- 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.

- (We used "long division" to briefly describe the method used by
Michaela, but

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.

- For comparing 7/8 to 2/3, PSSM recommends the use of physical "fraction strips", never mentioning the concept of converting to a common denominator. -PSSM, Page 216
- For dividing 5 by 3/4 they recommend the method of "repeated subtraction," after first suggesting (see the following quote) that "invert and multiply" is too difficult for today's kids.
- How about 3/4 divided by 5 using repeated subtraction? Do they expect that kids will find it easy to use repeated subtraction to show that 9/11 divided by 3/121 equals 33? No, they will tell you that these are unreasonable divisions, and 11 and 121 are unreasonable denominators (see Connecticut). They say that students "need to see and explore a variety of models of fractions, focusing primarily on familiar fractions such as halves, thirds, fourths, fifths, sixths, eighths, and tenths." -PSSM, Page 150
__A comment from Professor Richard Askey__: "Given that the authors had a very nice chapter on this topic in Liping Ma's book with varied word problems and comments from teachers about such things as objecting to using 1 3/4 divided by 1/2 to see if students understood division of fractions since this is so easy to do without understanding how to divide fractions, I find it shocking that successive subtraction is pushed as the way to do division of fractions, and the final step when successive subtraction does not work is just 1/4 divided by 1/2. Even that is not adequately explained, it is just done. As the Chinese teacher suggested, this is too easy to see if division of fractions is understood or not."- Note: For the 5 divided by 3/4 problem discussed above in PSSM, the final step, 1/2 divided by 3/4, is not explained. They just say that 2/3 is left after 6 subtractions of 3/4.
- How does the student actually carry out the 6 subtractions of 3/4? We are told "students can visualize repeatedly cutting off 3/4 yard of ribbon" from 5 yards of ribbon. One wonders if they use scissors to help them "visualize".

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 usePSSM 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.sophisticated argumentsinvolving 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

**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:

- Subtraction with regrouping
- Multidigit multiplication
- Dividing fractions

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

- 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 - Ma is referenced on page 17 of PSSM, where we are told that teachers who know "fractions can be understood as parts of a whole, the quotient of two integers, or a number on a line" have an understanding that may be characterized as 'profound understanding of fundamental mathematics' (Ma, 1999)."
- This is not an illustration of PUFM. It is an example of a basic learning expectation for all students.
- 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).__

- 4th graders will continue
**not**to be expected to demonstrate pencil-and-paper mastery of: - subtraction with regrouping.
- 6th graders will continue
**not**to be expected to demonstrate pencil-and-paper mastery of: - addition and subtraction of numbers greater than 10,000 or money amounts greater than $100;
- multiplication and division by 2-digit or larger factors or divisors;
- addition and subtraction of fractions with unlike denominators; and
- computation with non-money decimals.
- 8th graders will continue
**not**to be expected to to demonstrate pencil-and-paper mastery of: - addition and subtraction of numbers greater than 10,000 or money amounts greater than $100;
- addition and subtraction of fractions, except halves and thirds or when one denominator is a factor of the other; and
- division with fractions or mixed numbers.

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:

- pencil and paper multiplication problems with two-digit or larger factors (3 digits by 1 digit should be enough);
- paper and pencil division problems with two-digit or larger divisors (4 digits by 1 digit should be enough); and
- computation with fractions with unreasonable denominators like sevenths or 11ths (halves, fourths, eighths; thirds and sixths; fifths and tenths should be enough).

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 theThe NCTM received excellent input (see examples below), but ignored it.Principles and Standardswriting group in the course of its work. -PSSM, Page xv

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

- What is meant by "algorithmic thinking"?
- How should the Standards address the nature of algorithms in their more general mathematical context?
- How should the Standards address the matter of invented and standard algorithms for arithmetic computation?
- What is it about the nature of algorithms that might be important for children to learn?

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 273Kenneth 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:. . . 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

to work forguaranteedproblems of the type they deal with—deserves emphasis. -Howe, Page 275 (bold added)allWe 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 algorithm is also significant for later understanding of real numbers. -Howe, Page 275

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 1The 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

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 1Knowing And Teaching Elementary Mathematics, By Richard Askey, John Boscom Professor of Mathematics, University of Wisconsin-Madison"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

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.

- As the word 'understanding' continues to be bandied about loosely in the
debates over math education, this book provides a much-needed grounding. It
disabuses people of the notion that elementary school mathematics is simple—or
easy to teach. It cautions us, as Ma says in her conclusion, that
'

(Bold emphasis added)