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9Teaching School Mathematics: Pre-AlgebraTeaching School Mathematics: Pre-Algebra

Hung-Hsi Wu

Teaching School Mathematics: Pre-Algebra

https://doi.org/10.1090//mbk/098

A M E R I C A N M A T H E M A T I C A L S O C I E T Y

Teaching School Mathematics: Pre-Algebra

Hung-Hsi Wu

Department of MathematicsUniversity of California, Berkeley

P r o v i d e n c e , R h o d e I s l a n d

2010 Mathematics Subject Classification. Primary 97-01, 00-01, 97F40, 97F80, 97K50,97G50, 97G40, 97G30.

For additional information and updates on this book, visitwww.ams.org/bookpages/mbk-98

Library of Congress Cataloging-in-Publication Data

Names: Wu, Hongxi, 1940-Title: Teaching school mathematics. Pre-algebra / Hung-Hsi Wu.Description: Providence, Rhode Island : American Mathematical Society, [2016] | Audience:

Grades 6 to 8.- | Includes bibliographical references.Identifiers: LCCN 2016000117 | ISBN 9781470427207 (alk. paper)Subjects: LCSH: Mathematics–Textbooks. | Mathematics–Study and teaching (Elementary) |

Mathematics–Study and teaching (Middle school) | AMS: Mathematics education – Instruc-tional exposition (textbooks, tutorial papers, etc.). msc | General – Instructional exposition(textbooks, tutorial papers, etc.). msc | Mathematics education –Arithmetic, number theory –Integers, rational numbers. msc | Mathematics education –Arithmetic, number theory –Ratioand proportion, percentages. msc | Mathematics education –Combinatorics, graph theory,probability theory, statistics –Probability theory. msc | Mathematics education –Geometry –Transformation geometry. msc | Mathematics education –Geometry –Plane and solid geome-try. msc | Mathematics education –Geometry –Areas and volumes. msc

Classification: LCC QA43 .W8 2016 | DDC 512.9071/2–dc23 LC record available at http://lccn.loc.gov/2016000117

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©∞ The paper used in this book is acid-free and falls within the guidelinesestablished to ensure permanence and durability.

Visit the AMS home page at http://www.ams.org/

10 9 8 7 6 5 4 3 2 1 21 20 19 18 17 16

Dedicated to the memory of

Wolfgang and Leo

Contents

Chapters in the Companion Volume ix

Preface xi

Suggestions on How to Read This Volume xxi

Chapter 1. Fractions 11.1. Definition of a fraction 31.2. Fractions as division 211.3. Equivalent fractions 271.4. Adding and subtracting fractions 431.5. Multiplying fractions 561.6. Dividing fractions 701.7. Complex fractions 881.8. FASM 951.9. Percent, ratio, and rate problems 991.10. Finite probability 1211.11. Appendix 143

Chapter 2. Rational Numbers 1452.1. The two-sided number line 1452.2. Adding rational numbers 1472.3. Subtracting rational numbers 1602.4. Multiplying rational numbers 1642.5. Dividing rational numbers 1742.6. Comparing rational numbers 1832.7. FASM, revisited 197

Chapter 3. The Euclidean Algorithm 2033.1. The reduced form of a fraction 2033.2. The Fundamental Theorem of Arithmetic 219

Chapter 4. Experimental Geometry 2294.1. Overview 2294.2. Freehand drawing 2364.3. Constructions using tools 2394.4. The basic isometries 2524.5. Congruence 287

vii

viii CONTENTS

4.6. Dilation 3014.7. Similarity 320

Chapter 5. Length, Area, and Volume 3395.1. The concept of geometric measurement 3405.2. Length 3485.3. Area 3565.4. Volumes of cylinders and cones 376

Bibliography 381

Chapters in the Companion Volume

Teaching School Mathematics: Algebra ([Wu-Alg])

Chapter 1: Symbolic ExpressionsChapter 2: Translation of Verbal Information into SymbolsChapter 3: Linear Equations in One VariableChapter 4: Linear Equations in Two Variables and Their GraphsChapter 5: Simultaneous Linear EquationsChapter 6: Functions and Their GraphsChapter 7: Linear Functions and Proportional ReasoningChapter 8: Linear Inequalities and Their GraphsChapter 9: ExponentsChapter 10: Quadratic Functions and Their Graphs

ix

x CHAPTERS IN THE COMPANION VOLUME

Structure of the chapters in this volume ([PA]) and [Wu-Alg] ([A])

[A]Chapter 10

[A]Chapter 7

[A]Chapter 5

[A]Chapter 6

[A]Chapter 4

[A]Chapter 3

[A]Chapter 2

[A]Chapter 1

[PA]Chapter 4

[PA]Chapter 2

[PA]Chapter 1

[PA]Chapter 3

[PA]Chapter 5

[A]Chapter 9 [A]Chapter 8

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Preface

Training has no shortcuts.

Golden State WarriorsRamp Run video,October 24, 2012

([GoldenState])

This volume and its companion volume—Teaching School Mathematics:Algebra ([Wu-Alg])—address the mathematics that is generally taught ingrades 5–9. They are not student texts, however, because they have beenwritten expressly for teachers, especially middle school teachers. Thesetwo volumes are designed not to show you how mathematics is really justcommon sense and lots of fun, but to help you teach the mathematics ofmiddle school in a way that meets the minimal standards of human com-munication. In other words, problems are solved without recourse to tricksor any ad hoc sleight-of-hand, every step is explained logically using onlyconcepts and skills already developed, and every concept is clearly definedso that no clever guessing is needed for its understanding. There may bean added bonus in that the mathematical development of these volumesparallels that of the Common Core State Standards for Mathematics ([CC-SSM]) for middle school.

These volumes differ from the usual presentations found in standardschool textbooks (and professional development materials as well) in sub-stantial ways. First and foremost, the presentations in the standard text-books, be they traditional or reform, are riddled with mathematical errors,thanks to Textbook School Mathematics (TSM).1 While the Table of Contentsbears a superficial resemblance to what you normally find in school text-books and other professional development materials, there are major dif-ferences in terms of precision, sequencing, and reasoning. It is hoped thatthese volumes will lead you to rethink some of this material even if youbelieve you already know it very well.

1This is the name given to the mathematics in almost all standard school mathematicstextbooks of roughly the past four decades. It is notable for being antithetical to the fiveprinciples listed on pages xv ff. A more elaborate discussion of TSM can be found in[Wu2013b] and [Wu2015].

xi

xii PREFACE

The first major departure from TSM in these volumes is the treatmentof fractions and rational numbers. Fractions (and rational numbers) arethe backbone of K–12 mathematics and are therefore the centerpiece ofnot only these two volumes, but also the other volumes written for teach-ers: [Wu2011a] and [Wu-HighSchool]. Contrary to the prevailing norm inmathematics education, these volumes will ask you to spread the messagethat:

(1) Fractions are numbers that you can compare to see which is big-ger, and can add, subtract, multiply, and divide.

(2) The number line is home for all (real) numbers, including wholenumbers, fractions, and rational numbers.

(3) Fractions of a fixed denominator, when viewed as multiples of thecorresponding unit fraction, are just like whole numbers, at leastin terms of addition and subtraction.

(4) Students should get to know what a fraction is and what it meansto add, subtract, multiply, and divide fractions before they per-form the formal procedures of fraction arithmetic.

(5) The least common denominator is not needed for adding fractions,and there is no compelling mathematical reason to insist that frac-tions be always reduced to lowest terms.

(6) Finite decimals are a special class of fractions.(7) Everything we need to know about fractions, including multipli-

cation and division, can be explained using the definition of afraction as a point on the number line.

(These emphases were first put forth in [Wu1998], and can be found incomplete detail in [Wu2002]; they are also present in [Jensen].)

A second major departure lies in the heavy emphasis placed on geom-etry in the middle school curriculum, especially on giving precise defini-tions for the concepts of congruence and similarity. According to TSM,congruence means same size and same shape and similarity means same shapebut not necessarily the same size. As mathematics, this is unacceptable be-cause “same size” and “same shape” are words that can mean differentthings to different people, whereas mathematics only deals with clear andunambiguous information. What these volumes promote is a different ap-proach to the teaching of these concepts. Take congruence, for example.First make sure that you know what translations, reflections, and rota-tions are, then devise hands-on activities for your students to familiarizethemselves with these transformations, and, finally, teach them that, bydefinition, two geometric figures are congruent if one figure can be carried ontothe other by the use of a finite number of translations, reflections, and rotations.Conceptually the same thing can be said about similarity. These volumeswill help you acquire the requisite knowledge you need to teach congru-ence and similarity differently—and better.

PREFACE xiii

The heavy emphasis on geometry all through both volumes is moti-vated by the fact that—contrary to what TSM would have you believe—familiarity with similar triangles is absolutely crucial to the learning of lin-ear equations in algebra, particularly the concept of the slope of a line (seeChapter 3 of [NMP2], [Wu2010b], and [Wu2010c]). Students’ understand-ing of the concept of slope is a main stumbling block in beginning algebra(see, e.g., [Postelnicu]), and one of the contributions of these volumes isa different approach to the definition of slope that is more intuitive andmakes entirely obvious why certain lines have negative slope (see Section4.3 in [Wu-Alg]).

While the geometric topics taken up are, with but one exception,2

what one normally finds in the standard middle school curriculum—translations, reflections, rotations, congruence, length, area, volume, etc.—they are not taken up as fun, optional activities. Rather, these are topicsthat are essential for the learning of algebra and, to that end, are put touse in [Wu-Alg] for substantive logical reasoning in the discussion of thegraphs of linear equations, linear functions, linear inequalities, and qua-dratic functions. For example, having a correct definition of the slope ofa line makes it possible for teachers to explain, and for students to under-stand (rather than merely memorize), why the graph of a linear equationax + by = c is a line (see Section 4.4 in [Wu-Alg]). The absence of thisreasoning in TSM has made the writing down of the equation of a line thatsatisfies certain geometric data a fearsome task to many students of alge-bra. But teachers who have been exposed to this reasoning will begin tosee how they might teach the graphing of linear equations differently andliberate their students from this fear, because reasoning can now replacerote memorization.

Beyond the implications for the teaching of algebra, the other reasonfor the emphasis on geometry in the middle school curriculum is thattranslations, rotations, reflections, and dilations provide a much more ac-cessible introduction to the staple of a rigorous high school course on ge-ometry: the study of triangles and circles (cf. Volumes I and II of [Wu-HighSchool]). Because the learning of these transformations can be mademore accessible and greatly expedited through the use of hands-on geo-metric experiments, the hands-on experiences serve to demystify congru-ence and similarity for students. At a time when the school geometrycurriculum is beset by issues of fragmentation (because of the disconnectbetween middle school geometry and high school geometry) and meaning-less abstraction (as a result of the rote application of the axiomatic methodin a school setting), the middle course offered in these volumes is one po-tential solution to this pressing curricular problem. For a more detaileddiscussion of these ideas, see Section 4.1 on page 229.

2The one exception is the concept of dilation.

xiv PREFACE

The final major departure from TSM in these volumes is the emphasisput on the careful use of symbols. The concept of a “variable” is at presentthe scourge of middle school mathematics that bars any meaningful entryinto algebra. In mathematics, “variable” is no more than an informal pieceof terminology that serves to remind us of an element in the domain of afunction. Yet in TSM and the education literature, “variable” has been ele-vated to the status of a mathematical concept. The inevitable result of suchan aberration is to make introductory algebra unlearnable. The whole ofthe companion volume [Wu-Alg] will testify to the fact that when carefulattention is given to the correct use of symbols, rather than to the contor-tions involved in trying to make sense of “variable”, every foundationalconcept and skill in introductory algebra (what is an equation? what doesit mean to solve an equation? what is an expression? etc.) gains in clarityand conceptual simplicity, and algebra becomes once again a potentiallylearnable subject.

Although these two volumes (an expansion of [Wu2010b] and[Wu2010c]) have been used in my professional development institutessince 2006, it has been difficult to convince teachers to put such a mathe-matical development directly to use in their classrooms. Their reluctance isentirely understandable because doing so would entail the need to developnew classroom lessons—and probably new curricular units—on their own.It would also require them to teach against the existing curriculum of TSM.For example, according to TSM, fractions are best understood through theuse of analogies and metaphors (compare the critique in pages 34–39 of[Wu2008]), the concept of a “variable” is central to middle school math-ematics (page 102 of [NCTM]), and similar triangles are irrelevant to thelearning of school algebra (look up almost any school algebra textbookin K–12 in the past four decades). It is unfair to ask teachers to single-handedly defy such an entrenched tradition.

This situation has changed somewhat with the advent of the Com-mon Core State Standards for Mathematics (CCSSM) (see [CCSSM]). TheCCSSM have come to substantial agreement with the main advocacies ofthese volumes,3 especially the three major departures from TSM men-tioned above. A recent article in Education Week ([Heitin]) indicates that,perhaps, educators have finally come around to embracing the main em-phases on the teaching of fractions in (1)–(7) above (one can gain a littlehistorical perspective on this issue by reading Chapter 24 of [Wu2011a]).It should now be easier to convince teachers to learn and apply the con-tent of these volumes (and to convince their administrators to allow themto do so) because the CCSSM are now being implemented in most states.This fact acquires additional significance because on the one hand, schooltextbooks in general have not risen to the challenge of the CCSSM as of

3The document [Wu2010b] is the same document as the one cited as “Wu, H. ‘LectureNotes for the 2009 Pre-Algebra Institute,’ September 15, 2009” on page 92 of [CCSSM].

PREFACE xv

November 2015, and on the other, there seems to be no other completemathematical exposition of middle school mathematics that is consistentwith the CCSSM—this is especially true for fractions, negative numbers,and geometry. My hope is that these volumes can double as a stopgapmeasure at a time when the implementation of the CCSSM seems not toosure of its mathematical footing.

An original impetus for the writing of these volumes was to help solveour nation’s severe mathematics education crisis.4 Back in 2004 whenthis work was first conceived, the CCSSM did not exist, but the glaringdefects of TSM could not be ignored. There are good reasons to believethat the writing of the CCSSM was inspired by this same crisis. It is finallytime to banish from schools the jumbled, chaotic, and even downrightanti-mathematical presentations that characterize and pervade TSM. Tothis end, the present volumes strive to improve mathematics teachingby emphasizing, throughout, the following five fundamental principles(compare [Wu2011b]):

(I) Precise definitions are essential. In mathematics, precise definitionsare the bedrock on which all logical reasoning rests, because mathematicsdoes not deal with vaguely conceived notions. Yet definitions are lookedupon with something close to disdain by most teachers (and students) asjust “more things to memorize”. Such a fundamental misconception ofthe basic structure of mathematics can only come from the TSM we all re-member from our own schooling and now teach again to our students, andfrom the flawed professional development we provide for our teachers. Inthese volumes, we will respect this fundamental characteristic of mathe-matics by offering—and employing—precise definitions for every concept,including those that are commonly used, yet remain undefined, in TSM:fractions, decimals, sum of fractions, product of fractions, division of fractions,ratio, percent, rate, equation, congruence, similarity, slope of a line, graph of aninequality, polygon, length, area, etc.

(II) Every statement must be supported by reasoning. There are no unex-plained assertions in these volumes.5 If something is true, a logical ex-planation will be given. Although it takes some effort to learn the logicallanguage used in mathematical reasoning, in the long run the presence ofreasoning in all we do has the advantage of disarming disbelief and remov-ing the stress of learning-by-rote. It also has the salutary effect of puttingthe learner and the teacher on the same footing, because the ultimate ar-biter of truth will no longer be the teacher’s or the textbook’s authority,but the compelling rigor of the reasoning.

4See, for example, [Askey], [RAGS], and [NMP1].5Except those few explicitly designated as such, because their proofs require advanced

mathematics.

xvi PREFACE

(III) Mathematical statements are precise. In mathematics, there is no roomfor imprecision because imprecision leads to misunderstanding and there-fore errors. TSM, however, is rife with imprecision, saying things suchas “the pizza is the whole” in the study of fractions. This leads to mis-conceptions about the “whole” being a shape (the circle), whereas what ismeant mathematically is that the whole is the area of the pizza. TSM alsodefines percent to be out of a hundred. This then leaves students confusedas to whether percent is an “action” or a number. If it is an “action”, howdoes one add and divide “actions”, and if it is a number, what kind ofnumber is it? It is difficult to imagine how mathematics learning can takeplace when learners’ minds are beset by such confusion. Another exampleis TSM’s claim that “multiplication and division are inverse operations;they undo each other”. But given two numbers such as 2 and 3, we have2 × 3 = 6 and 2 ÷ 3 = 2

3 . TSM does not explain in which way 6 and 23

undo each other. What is meant is that if we fix a number k ( �= 0), then theoperation of multiplying a given number by k followed by the operation ofdividing the resulting number by k leaves the given number unchanged;in this sense, multiplication and division indeed undo each other. It wouldseem, however, that even this much precision is unattainable by TSM. Thisis another reason why TSM is unlearnable.

This lack of precision is by no means limited to elementary schoolmathematics; it pervades the K–12 curriculum. On the high school level,for example, the definition that 3−x = 1/3x is too often offered amidsta flurry of heuristic arguments that leave the readers with the impressionthat the equality 3−x = 1/3x has been proved. Such persistent ambiguitiesconsequently leave many students as well as teachers confused about thedifference between a definition and a theorem.

(IV) Mathematics is coherent. The concept of mathematical coherence isoften brought up in educational discussions nowadays, but it is not some-thing that can be understood through verbal descriptions any more thanthe transcendental serenity of the adagio in the Schubert C major quintetcan be appreciated through the reading of an essay praising its beauty.Very crudely speaking, the coherence of mathematics refers to the fact thatthe body of knowledge that is mathematics has a tightly-knit structure,but the only way one can get to know and appreciate this structure isby wading into its details. For example, the concept of similarity in Sec-tion 4.7 on page 320 relies on a knowledge of multiplying and dividingfractions (Sections 1.5 and 1.6 on pages 56 and 70, respectively) and con-gruence (Section 4.5 on page 287), and is itself used in a crucial way forthe definition of slope (Section 4.3 in [Wu-Alg]). Another example is theomnipresence of the theorem on equivalent fractions in the discussion ofalmost every topic in fractions, when TSM would have you believe thatit is only useful for simplifying fractions. Yet another example is the keyrole played by congruence not only in the definition of similarity (Section4.7 on page 320) but also in the considerations of length, area, and volume

PREFACE xvii

(see Chapter 5). As a final example, you will notice that the division ofwhole numbers, the division of fractions (Section 1.6 on page 70), and thedivision of rational numbers (Section 2.5 on page 174) are conceptuallyidentical.

The coherence of mathematics makes mathematics more teachable andmore learnable. This can be easily understood by an analogy: whereasone can pore over a page from a phone book without any recollection ofwhat has been read afterwards, almost all readers have vivid memories ofDon Quixote—all one thousand pages of it—even after only one reading,because it tells a coherent story.

Although coherence is difficult to describe, the lack of coherence can bemore easily illustrated. A striking example of the failure of coherence inTSM is the common explanation of the theorem on equivalent fractions,which states that m

n = kmkn for all fractions m

n and for all positive integers k.TSM would have you believe that this is true because

mn

= 1 × mn

=kk× m

n=

kmkn

.

However, the last step depends on knowing how to multiply fractions, andthe multiplication of fractions is a topic that comes late in the developmentof the subject.6 When the reasoning for the basic theorem in fractions—the theorem on equivalent fractions—is given in terms of something morecomplex and, in any case, not yet available, how can we expect students tolearn? Unfortunately, such subversions of logic abound in TSM.

(V) Mathematics is purposeful. Mathematics is goal-oriented, andevery concept or skill in the standard curriculum must be there for apurpose. Teachers who recognize the purposefulness of mathematics gainan extra tool for making their lessons more compelling and, therefore,more learnable. When congruence and similarity are taught with noapparent purpose except to do “fun activities”,7 students lose sight of themathematics and may wonder why they are required to learn it. However,as noted above, the concept of congruence lies behind the concept ofsimilarity, and both are needed to make sense of basic issues in algebrasuch as linear equations of two variables and their graphs, e.g., why is thegraph of such an equation a (straight) line? Students are more likely tofeel motivated to learn if presented with a curriculum that actually offersexplanations of why its basic facts are worth learning.

6Multiplication is the most subtle among the four arithmetic operations on fractions.Its definition is nontrivial; the proof of the product formula is sophisticated; and its rela-tionship with the area of a rectangle (with fractional sides) is subtle. See Section 1.5 onpage 56.

7This has been happening all too often lately as a result of the misunderstanding ofthe CCSSM propagated by people immersed in TSM.

xviii PREFACE

Middle school mathematics is the bridge that leads from fairly concreteconcepts about numbers in elementary school to more abstract conceptsin algebra, geometry, and trigonometry in high school. Our nation’scurriculum is traditionally weak in middle school; one can almost saythat it has been delinquent in its failure to provide careful guidance forstudents’ transition from the concrete to the abstract. The teaching of TSMhas become the norm in those years. Instead of giving precise instructionon the correct use of symbols and explaining the need for the idea ofgenerality in students’ next step on their mathematical journey, TSM harpson the alleged profundity of the fictitious concept of a “variable”; insteadof guiding students’ tentative first steps to think abstractly about negativenumbers, TSM redirects them to replace abstract thinking by analogiesand heuristic patterns. By contrast, this volume and its companionvolume [Wu-Alg] take this bridge seriously. They confront the necessaryabstractions without compromise, but they do so by building on thefoundation of elementary school mathematics (cf. [Wu2011a]). I hopethese volumes will initiate change by making you more aware of theoverriding importance of this bridge in students’ mathematics learningtrajectory. Ultimately, the goal of these volumes is to help you teach yourstudents better.

Acknowledgements. This volume and its companion volume [Wu-Alg] evolved from the lecture notes ([Wu2010b] and [Wu2010c]) for thePre-Algebra and Algebra summer institutes that I used to teach to mid-dle school mathematics teachers from 2004 to 2013. My ideas on profes-sional development for K–12 mathematics teachers were derived from twosources: my understanding as a professional mathematician of the min-imum requirements of mathematics (see the five fundamental principleson pages xv ff.) and the blatant corrosive effects of TSM on the teachingand learning of mathematics. Those summer institutes therefore placed aspecial emphasis on improving teachers’ content knowledge. I would nothave had the opportunity to try out these ideas on teachers but for thegenerous financial support from 2004 to 2006 by the Los Angeles CountyOffice of Education (LACOE), and from 2007 to 2013 by the S. D. Bech-tel, Jr. Foundation. Because of the difficulty I have had with funding bygovernment agencies—they did not (and perhaps still do not) consider thekind of content-based professional development I insist on to be worthyof support—my debt to Henry Mothner and Tim Murphy of LACOE andStephen D. Bechtel, Jr. is enormous.

Through the years, I have benefited from the help of many dedicatedteachers; to Bob LeBoeuf, Monique Maynard, Marlene Wilson, and BettyZamudio, I owe the corrections of a large number of linguistic infelicitiesand typos, among other things. Winnie Gilbert, Stefanie Hassan, and SunilKoswatta were my assistants in the professional development institutes,and their comments on the daily lectures of the institutes could not help

PREFACE xix

but leave their mark on these volumes. In addition, Sunil created someanimations (referenced in Chapters 2 and 4) at my request. Phil Daro gra-ciously shared with me his insight on how to communicate with teachers.Sergei Gelfand made editorial suggestions on these volumes—includingtheir titles—that left an indelible imprint on their looks as well as theiruser-friendliness. R. A. Askey read through a late draft with greater carethan I had imagined possible, and he suggested many improvements aswell as corrections. I shudder to think what these volumes would havebeen like had he not caught those errors. Finally, Larry Francis helpedme in multiple ways. He created animations for me that can be found inChapter 4. He is also the only person who has read almost as many draftsas I have written. (He claimed to have read twenty-seven, but I think heoverestimated it!) He met numerous last minute requests with unfailinggood humor, and he never ceased to be supportive; more importantly, heoffered many fruitful corrections and suggestions.

To all of them, it gives me great pleasure to express my heartfelt thanks.Hung-Hsi Wu

Berkeley, CaliforniaMarch 15, 2016

Suggestions on How to Read This Volume

The major conclusions in this book, as in all mathematics books, aresummarized into theorems; depending on the author’s (and other mathe-maticians’) whims, theorems are sometimes called propositions, lemmas,or corollaries as a way of indicating which theorems are deemed more im-portant than others (note that a formula or an algorithm is just a theorem).This idiosyncratic classification of theorems started with Euclid around 300B.C., and it is too late to change now. The main concepts of mathematicsare codified into definitions. Definitions are set in boldface in this bookwhen they appear for the first time. A few truly basic definitions are evenindividually displayed in a separate paragraph, but most of the definitionsare embedded in the text itself. Be sure to watch out for them.

The statements of the theorems as well as their proofs depend on thedefinitions, and proofs (= reasoning) are the guts of mathematics.

A preliminary suggestion to help you master the content of this bookis for you to

copy out the statements of every definition, theorem,proposition, lemma, and corollary, along with page refer-ences so that they can be examined in detail if necessary,

and also to

summarize the main idea of each proof.

These are good study habits. When it is your turn to teach your students,be sure to pass on these suggestions to them. A further suggestion is thatyou might consider posting some of these theorems and definitions in yourclassroom.

You should accept as agiven that mathematicsbooks make for exceedinglyslow reading.

You should also be aware that readingmathematics is not the same as reading agossip magazine. You can probably flipthrough such a magazine in an hour, if notless. But in this book, there will be manypassages that require careful reading andre-reading, perhaps many times. I cannotsingle out those passages for you because they will be different for dif-ferent people. We do not all learn the same way. What is true under allcircumstances is that you should accept as a given that mathematics booksmake for exceedingly slow reading. I learned this very early in my career.

xxi

xxii SUGGESTIONS ON HOW TO READ THIS VOLUME

On my very first day as a graduate student many years ago, a professor,who was eventually to become my thesis advisor, was lecturing on a par-ticular theorem in a newly published volume. He mentioned casually thatin the proof he was going to present, there were two lines in that book thattook him fourteen hours to understand and he was going to tell us whathe found out in those long hours. That comment greatly emboldened menot to be afraid to spend a lot of time on any passage in my own reading.

If you ever get stuck in any passage of this book, take heart, becausethat is nothing but par for the course.

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[Beck-Bleicher-Crowe] A. Beck, M. N. Bleicher, and D. W. Crowe, Excursions into Mathe-matics, Worth Publishers, 1969, or A K Peters, Ltd., 2000.

[Behr-Lesh-Post-Silver] M. Behr, R. Lesh, T. Post, and E. Silver, Rational Number Con-cepts. In R. Lesh and M. Landau (Eds.), Acquisition of Mathematics Concepts andProcesses, Academic Press, New York, NY, 1983 (pp. 91-125).

[Birkhoff-Mac Lane] G. Birkhoff and S. Mac Lane, A survey of Modern Algebra, 4th Edition,MacMillan, NY, 1977.

[Carpenter et al.] T. P. Carpenter, M. L. Franke, and L. Levi. Thinking mathematically: Inte-grating algebra and arithmetic in elementary school, Heinemann, Portsmouth, NH,2003.

[CCSSM] Common Core State Standards for Mathematics, June, 2010. Retrieved fromhttp://www.corestandards.org/Math/

[Chung] K. L. Chung, Elementary Probability Theory with Stochastic Processes, Third edition,Springer-Verlag, New York-Heidelberg-Berlin, 1979.

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[Euclid] Euclid, The Thirteen Books of the Elements, Volumes 2 and 3, T. L. Heath, transl.,Dover Publications, New York, 1956.

[Gauss] C. F. Gauss, Disquisitiones Arithmeticae (Second, corrected edition), A. A. Clarke,translator, Springer, New York-Heidelberg, 1986.

[Ginsburg] J. Ginsburg, On the early history of the decimal point, American MathematicalMonthly, 35 (1928), 347–349.

[GoldenState] Golden State Warriors Ramp Run.https://www.youtube.com/watch?v=q3mJ0_C3p9s

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