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On April 12, 2000, The National Council of Teachers of Mathematics (NCTM)\u00a0 released Principles and Standards for School Mathematics<\/a> (PSSM<\/b>), a 402 page revision of\u00a0 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.”\u00a0 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.”<\/span><\/p>\n 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<\/u>) under “News and Hot Topics” at\u00a0 NCTM Speaks Out: Setting the Record Straight about changes in Mathematics Education<\/a>.<\/span><\/p>\n 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of how to use technology to arrive meaningfully at solutions to problems instead of endless attention to increasingly outdated computational tedium.\u00a0\u00a0 -NCTM,\u00a0 Commonsense Facts to Clear the Air<\/u><\/span><\/p><\/blockquote>\n It’s clear that The New York Times was fed misleading NCTM propaganda, perhaps designed to placate “math wars” opponents.\u00a0 Not surprisingly, we will show here that the NCTM has not rediscovered arithmetic.\u00a0 Similar to the original NCTM Standards, PSSM is vague about the major components of arithmetic mastery:<\/span><\/p>\n The NCTM has toned down the constructivist language, but they still stress content-independent “process skills” and student-centered “discovery learning”.\u00a0 Similar to the NCTM Standards, PSSM emphasizes manipulatives, calculator skills, student-invented methods, and simple-case methods.<\/span><\/p>\n Although PSSM contains five “Connections” sections,\u00a0 there continues to be no acknowledgement of the vertically-structured nature of\u00a0 mathematics.\u00a0 Mastery of math requires a step-by-step build up (in the brain) of specific content knowledge.\u00a0 PSSM omits this aspect of the “connections” within mathematics.\u00a0 The idea conveyed by the following example is not found in PSSM.<\/u><\/span><\/p>\n Exampl<\/u>e: Migrating up the math learning curve<\/b><\/span><\/p>\n 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.<\/span><\/p>\n 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.\u00a0 Although rarely the preferred method, intentional memorization is sometimes the most efficient approach.\u00a0 The first objective is to get it into the brain!\u00a0 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.”<\/span><\/p>\n 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<\/u> (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.<\/b>\u00a0 The voices of reason have been largely silenced!\u00a0 Here is an example of the silencing.<\/span><\/p>\n<\/a> 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.\u00a0 -PSSM Draft, Page 43<\/span><\/p>\n Technology should not be used as a replacement for basic understandings and intuitions; rather it should be used to foster those understandings and intuitions.\u00a0 -PSSM, page 24<\/span><\/p><\/blockquote>\n<\/a> 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.<\/span><\/p>\n 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.”<\/span><\/p>\n<\/a> 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.\u00a0 -PSSM Draft, Page 51<\/span><\/p>\n A certain amount of practice is necessary to develop fluency with both basic fact recall and computation strategies for multi-digit numbers.\u00a0 Anderson, Reder, and Simon (1996) point out that practice is clearly essential for acquiring cognitive skills of almost any kind. -PSSM Draft, Page 114<\/span><\/p><\/blockquote>\n<\/a> Fluency with basic addition and subtraction number combinations is a goal for pre-K-2 years.\u00a0 By fluency<\/i> we mean that students are able to compute efficiently and accurately with single-digit numbers.\u00a0\u00a0 -PSSM, Page 84<\/span><\/p>\n When students leave grade 5, they should be able to . . . efficiently recall or derive the basic number combinations for each operation.\u00a0 -PSSM, Page 149<\/span><\/p>\n 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<\/span><\/p>\n 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<\/span><\/p><\/blockquote>\n The changes may be subtle, but notice that fluency<\/u> with basic number facts is defined as the ability to “compute efficiently and accurately.”\u00a0\u00a0 It does not mean the ability to instantly recall.\u00a0 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.\u00a0 Basic number fact “thinking strategies” appear to be preferred by the writers of PSSM.\u00a0 They give grudging admission that memorization may be necessary.\u00a0 There is some discussion of activities to teach fact relationships, but there is no discussion of mastery activities to facilitate fact memorization.<\/span><\/p>\n<\/a> 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\u00a0 relevant quotes from PSSM:<\/span><\/p>\n In the past, common school practice has been to present a single algorithm for each operation.\u00a0 However, more than one efficient and accurate computational algorithm exists for each arithmetic operation.\u00a0\u00a0 In addition, if given the opportunity, students naturally invent methods to compute that make sense to them.\u00a0 -PSSM, Page 153<\/span><\/p>\n 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).\u00a0 For example, many students and adults use multiplication to solve division problems or add starting with the largest place rather than with the smallest.\u00a0 The conventional algorithms for multiplication and division should be investigated<\/b> in grades 3 – 5 as one<\/b> efficient way<\/b> to calculate. -PSSM, Page 155\u00a0\u00a0 (bold emphasis added)<\/span><\/p>\n Students\u2019 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.\u00a0 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.\u00a0 -PSSM, Page 220<\/span><\/p>\n This last quote contains a hint of truth.\u00a0 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.\u00a0 This is the essence of the lesson method used in Japan.\u00a0 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.”<\/span><\/p>\n<\/a> On page 32 of PSSM,\u00a0 the term “computational fluency” is defined as “having and using efficient and accurate methods for computing”.\u00a0 Later on the same page, we are told that students should “see the usefulness of methods that are efficient, accurate, and general.”\u00a0 On page 87\u00a0 we are told\u00a0 “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.”<\/span><\/p>\n Has the definition of computational fluency been (appropriately) expanded to include “general”?\u00a0\u00a0 No, the original definition of computational fluency, including only efficient and accurate,\u00a0 is restated on pages 79 and 153.\u00a0\u00a0 It appears that some PSSM writers recognized that all three characteristics contribute to the power of the standard algorithms of arithmetic.\u00a0 But the standard algorithms are not mentioned in this context (page 87).\u00a0 Instead, on this page we are advised:\u00a0 “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.”<\/span><\/p>\n<\/a> Within the five “Number and Operations” sections, PSSM includes (only) three illustrations of\u00a0 multidigit computation. All are “student-invented” strategies.\u00a0 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.”\u00a0 There is no discussion of the accuracy, efficiency or generality of any method found in these three illustrations<\/u>. Apparently those who wrote about “computational fluency” failed to communicate with those who developed the examples.<\/span><\/p>\n In this last illustration, we again we have a hint of the lesson method used in Japan.\u00a0 But we learn that Mrs. Sparks objective is to “help the students understand, explain, and justify their computational strategies,”\u00a0 rather than working to achieve closure by\u00a0 connecting the two methods and teaching long division, emphasizing the efficiency and generality of the long division algorithm.<\/span><\/p>\n<\/a> 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.\u00a0 The phrase “common denominator” is not found in PSSM<\/u>.<\/span><\/p>\n 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.”\u00a0 – PSSM, Page 214.<\/span><\/p>\n PSSM’s treatment of fractions offers just two illustrations, one for comparing fractions,\u00a0 and the other for dividing fractions.\u00a0 Both are simple-case methods, and neither is efficient or general.<\/span><\/p>\n The division of fractions has traditionally been quite vexing for students.\u00a0 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.\u00a0 Some students forgot which number is to be inverted, and others are confused about when it is appropriate to apply the procedure.\u00a0 A common way of formally justifying the “invert and multiply” procedure is to use sophisticated arguments<\/u> involving the manipulation of algebraic rational expressions\u2014arguments beyond the reach of many middle-grade students.\u00a0 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.”\u00a0 -PSSM, page 219\u00a0\u00a0 (underline added)<\/span><\/p>\n For the sophisticated arguments<\/u>, see pages 2-3 in Basic Skills Versus Conceptual Understanding: A Bogus Dichotomy in Mathematics Education<\/a>, By H.Wu<\/span><\/p><\/blockquote>\n PSSM recommends that probability be covered in every grade, offering five PSSM sections on “Data Analysis and Probability”.\u00a0 The NCTM is not bothered by the fact that any meaningful discussion of elementary probability requires prior mastery of fractions.<\/span><\/p>\n Liping Ma’s Book: U.S. Elementary School Teachers Don’t Understand Arithmetic<\/b><\/span><\/p>\n 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.\u00a0 This is one message of the new book,\u00a0 Knowing and Teaching Elementary Mathematics<\/u> (KTEM) by Liping Ma.\u00a0 (Please see Roger Howe’s Review of Liping Ma’s Book<\/a>\u00a0 and\u00a0 Richard Askey’s Review of Liping Ma’s Book<\/a>).<\/span><\/p>\n U.S. teachers fared poorly when asked questions related to the teaching of:<\/span><\/p>\n Don’t blame the teachers!\u00a0 None of these topics appear in PSSM or the PSSM Draft<\/u>.\u00a0 The term “subtraction with regrouping (or renaming)” is never used.\u00a0 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.”\u00a0 The failure to cover division by fractions was discussed above.<\/span><\/p>\n Liping Ma’s book is referenced in two ways in PSSM.\u00a0 In each case her ideas have been misused:<\/span><\/p>\n\n
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Technology: PSSM Draft vs. PSSM<\/span><\/h4>\n
They Still Call It “Arithmetic”<\/span><\/h3>\n<\/a>
Mastery of Basic Facts or Derive Them When Needed?<\/span><\/h4>\n
Quotes From The PSSM Draft:\u00a0 Recall<\/u> Number Facts Quickly<\/span><\/h4>\n
Quotes From PSSM:\u00a0 Derive<\/u> Number Facts Quickly<\/span><\/h4>\n
Mastery of Standard Algorithms or Student-Invented Algorithms?<\/span><\/h4>\n
Efficient, Accurate, and (Possibly) General Methods<\/span><\/h4>\n
The Truth is in The Examples<\/span><\/h4>\n
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Mastery of Fractions or Simple-Case Methods for “Familiar Fractions”?<\/span><\/h4>\n
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\neasy to see if division of fractions is understood or not.”<\/span><\/p>\n\n
PSSM on “invert and multiply””<\/span><\/h4>\n
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