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Fehr, Howard F.Secondary School Mathematics Curriculum ImprovementStudy. Final Feport.Secondary Scnool MatheNatics Curriculum ImprovementStudy, New York, N.Y.Office of Education (DEEW), Washington, D.C. Pureauof Research.b?-7-0711Aug 70OFC-1-7-07C711-442023p.

IDRS Price MF-$0.6S FC-i3.2c2,*Curriculum Development, 4Ir.struction, MathematicsEducation, *Projects, 4'Se%:enclary School Mathematics

This rgcrt Ae planninj, 'riling,evaluation, and results and recommendations of the Secondary SchoolMathematics Cutriculua Improvement Study. This study was begun "toformulate and test a unified secondary school mathematics program(7-12) that will take capable students well into current collegiatemathematics:" and to "determine the education requited by teachers uhewill implement sucn a program." Also included in the report is ascheme for a taxonomy of objectives and course contents for each ofthe three courses developed (grades 7-e-9). Recommendations are thattne program he reexamined for possible adoption F.s a curriculum forall junior high school students, that the pteservice mathematic7--education or prospective junior high school teachers he reexamined,and that the unified approach developed for junior high school heextended throughout the senior high school. (FL)

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FINAL REPORTProject No. 7-0711

Contract No. OEC 1-7-070711-4420

SECONDARY SCHOOL MATHEM ATICS CURRICULUMIMPROVEMENT STUDY

August, 1970

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U.S. DEPARTMENT OF

HEALTH, EDUCATION, AND WELFARE

Office of EducationBureau of Research

FINAL REPORTProject No. 7-0711

Contract No. OEC 1-7-070711-4420

SECONDARY SCHOOL MATHEMATICS CURRICULUMIMPROVEMENT STUDY

Howard F. Fehr

Teachers College, Columbia University

New York, New York

August, 1970

The research reported herein was performed pursuantto a contract with the Office of Education, U.S. De-partment of Health, Education, and Welfare. Contrac-tors undertaking such projects under Government spon-sorship are encouraged to express freely their pro-fessional judgment in the conduct of the project.Points of view or opinions stated do not, therefore,necessarily represent official Office of Educationposition or policy.

U.S. DEPARTMENT OF

HEALTH, EDUCATION, AND WELFARE

Office of EducationBureau of Research

CONTENTS

PAGE

Acknowledgments iiiSummary 1

Introduction 2Planning the 7-12 Program 3Writing of Courses I, II, and IIIEducation of TeachersTeaching Courses I, II, and III 6Evaluation of Courses I, II, and III 6FUture ActivityConclusions and RecommendationsTable I (Scheme for Taxonomy of Objectives) 10Appendix A Course I Contents) 11Appendix B Course II Contents)

n 1.(Appendix C Coree ITT Contents) .

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AUCNOWLEDGMENTS

Planning, writing, and teaching of SSMCIS Courses I,II, and III was done with the cooperation of the schools andteachers in the Metropolitan New York Area and the followingconsultants:

Nicholas A. Branca, Teachers College, Columbia UniversityJohn Camp, Teachers College, Columbia UniversityGustave Choquet, Universite de Paris, FranceRay Cleveland, University of Calgary, CanadaJohn Downes, Emory UniversityHoward F. Fehr, Teachers College, Columbia UniversityJames Fey, University of MarylandAlan Gewirtz , City University of New YorkAbraham Glickoman, Bronx High School of Science, New YorkRichard Good, University of MarylandVincent Haag, Franklin And Marshall collegeThomas Hill, University of OklahomaPeter Hilton, rlorp141 UniversityJulius HIavaty, National Council of TAnenArs of MftthematiccMichael Hoban CPC, lone College, New YorkMeyer Jordan, City University of New YorkBurt Kaufman, Southern Illinois UniversityHoward Kellogg, Teachers College, Columbia UniversityJeremy Kilpatrick, Teachers College, Columbia UniversityErik Krintensen, Aarhus University, DenmarkHoward Levi, City University of New YorkEdgar R. Lorch, Columbia UniversityRichard C. Pocock, Houghton College, New YorkLennart Rade, Chalmers Institute of Technology, SwedenMyron F. Rosskopf, Teachers College, Columbia UniversityHarry Ruderman, Hunter College High School, New YorkHarry Sitomer, C.W. Post CollegeHans-Georg Steiner, University of Karlsruhe, GermanyMarshall H. Stone, University of MassachusettsStanley Taback, New York UniversityH. Laverne Thomas, State University College at Oneonta, N.Y.Albert W. Tucker, Princeton UniversityBruce Vogeli, Teachers College, Columbia UniversityLucian Wernick, Illinois Western State College

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SUMMARY

The Secondary School Mathematics Curriculum ImprovementStudy (SSMCIS) has two main objectives:

1) To formulate and test a unified secondaryschool mathematics program (7 - 12) that willtake capable students well into currentcollegiate mathematics;

2) To determine the education required byteachers who will implement such a program.

To inaugurate the study, leading United States and Europeanmathematicians and educators met in June 1966 to formulatea position paper suiting the aims and procedures of thestudy, to construct a flow charted analysis of the pro-posed 7 - 12 mathematics courses, and to make detailedrecommendations for the mathematical content of Course I.Using thin detailed syllabus as a guide, a team of eightmathematics educators wrote a textbook for Course I duringthat summer. Each chapter was written by one writer, re-viewed by the other writers and a consulting mathematician,and then revised for printing. Teachers' guides andsolutions to exercises were written and distributed tothe teachers.

In each subsequent year (1967, 1968) a two-week Juneworking conference was held to review and revise the pre-vious year's experimental text and to make specific re-commendations for the content and teaching; of a new course-Course II and Course III, respectively. As in the firstyear the writing team used the chapter guides dcvelopedby the June working conference to write the new texts forthe following year.

Six junior high schools in the metropolitan New York areahave participated in the experimental teaching of Courses I,II, and III from the initiation of the e;:periment. Each ofthese schools designated a team of two capable and interest-ed teachers who taught all the pilot classes using the ex-perimental textbooks. Each summer, while new materialswere being written, six weeks of instruction was given tothese teachers in preparation for teaching the new SSMCIScourse. This instruction included 50 hours in the funda-mental concepts underlying the unified mathematics programand 50 hourb in contemporary methods of teaching thoseconcepts.

The experimental teaching was evaluated in three ways. Thedirector and project staff members made frequent visits tothe classes for direct observation. The students weretested by examinations - prepared by the project staff - de-signed specifically to measure learning of important newconcepts introduced in the courses. Teachers, staff, andconsultants met at full day conferences to discuss progressand problems in the experimental teaching.

lir:sults of the experimentvl Lobehing have shown that thenew mathematics eourses, based on fundamental concepts and9tructures, give promive of veeting the expectations of theproposed six year pogvam,

Introduction

;11g the pA87, deend the United state: has been engaged insing the 01CMOWnrV nrti SL;.i001 mathematics

e Aculum - primarily by ';'0-(lainR the exinfInv traditionaleu) Modest reeimmenAntions of the eommissien onMallmatiee huode been le.rgely no.crred by curriculum andsyl)bun bodirF, hnd writrr of commercially producedtexepoks, lrpleehtatior, of thic program by the SMSGhats ad wile necep'.ence nnd nnr,siee exrerimental usethron^hote the eountry,

Throughon cl[ of ocr roforn frJovenrncs tho trbditionaidivinioe t,)t)uetiLe 1(11.o separate year:, ofarithmetle, liwobre, And ,,,eonetry hbc been maintained.Beyond introuetion c'r' new ennerW,e, little has been gainedin bringiog alivr. .eed s'uly int,o the tilgh schoolthrough more ficien. tvos,irofh of rgralizing the subjectmatter, and more radicn) reconTlendations for theimproveme;It of eeeoedary school education in mathematicshave been made 'ot.h 1.n this eourtrY, notebly by the UICSM,and in Europe, netmbly in nei!Aum, Switzerland, and Denmark.

What has beer, car: lel or reoon:Atruetion of the entirecurricolle(i fro n rlobnl r.oint of view - one whichelimineter the barriers 5opatatlng the nevem) branches ofmathematies qeld unifies the nebject through its generalconcepts (sets, opelltiens, maIpings, and relations) andbuilrha the fundamentel ntruetures of the nurber systems,algebra, and geometry (rower, :irge, fields, and vectorspaces). The effieiency ':sired by such organization shouldpermit introluetion Into the hiRli s.hool program of muchthat WOP previously consdeied undeegredunte mathematics.

t;

In September 1965, the Commissioner of Education, Departmentof Health, Education, and Welfare, Office of Education,approved for support for a period of 18 months the SecondarySchool Mathematics Curriculum Improvement Study (SSMCIS),an experimental study whose objective would be the construc-tion of a unified school mathematics curriculum for gradesseven through twelve.

In June, 1967, continuation of this support was grantedfor an additional 36-month period ending June 30, 1970.This is a report of the activities and findings of theSSMCIS during this latter period, covering the writing andteaching of experimental Courses II and III, the revisionand further, teaching of Courses I and II, and the finalrevision of Courses I and

Plare1:1212s2irsoIn June 1966 a group of nightprIn learl;ng United States andEuropean mathematicians and educators met for 20 days tooutline the scope and sequence of a six year unifiedsecondary school mathematics program. The first half ofthe conference was devoted to producing a complete flowcharted annlynis of the proposed course. When topicsplanned for the nerenth grade were expanded in workingpppers which outlined the mathematical content of eachtextbook chapter and made specific suggestions for writingand teaching these ideas,

Writing, of Courses I, II, and III

During July and August 1966, a team of eight mathematicaleducators wrote the textbook for Course I, using thesyllabus produced in June as a guide. Each textbookchapter was written by one writer, reproduced for reviewby the other writers and consulting mathematicians, andthen rewritten, incorporating the revewers, suggestions.Teachers' guides and solutions to exercises were written foreach chapter. These notes, mimeographed and distributed toteachers of experimental classes, included discussions offundamental mathematical ideas underlying each chapter,hints for possible class activity to accompany reading ofthe text, and suggested time allotment to the varioustopics, The Course I textbook was then published in threevolumes,

To initiate the detailed planning for Course II and forthe revisions of Course I, a pre-planning session was heldon March 11-12, 1967. The recommendations of this pre-planning group were considered in detail at a workingconference held in June, 1967, at which the full group of

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writers and consultants wan present. Thin group produced adetailed set of writing guidelines, which were then usedby the summer writing teams to produce the text materialsfor experimental Course II and to revise Course 1.

Beginning with a pre-planning meeting in December, 196y,and followed by a full working conference in June, 1968,a similar procedure wan used to plan and write experi-mental Course III, to rewrite Course II, and to finalizeCourse I. In addition, in order to make the fullestpossible use of the experience of two years of teachingCourse I, ten of the experimental teachers re-wrote andexpended the original teachers' commentaries for Course I,which had been originally written by the authors of thevarious chapters. These chapter-by-chapter commentarieswere then bound into a single volume and made availablefor public use along with the texts.

In 1969, the writing actIvitie:7 of the SSMCTS, sunoortedby the Office of Education, consisted of rewriting CourseIII and making final revisions in Course II. Again, asin the previous year, a'group of teachers of experimentalclasses worked during the summer, to rewrite the teacherscommentaries for Course II, which were then printed in asingle volume for use in conjunction with the texts,

The texts and Teachers Commentaries for ,.;ourses I and IIti each in two volumes) are now available for widespreadclassroom use, and are published and distributed by theTeachers College Press. When revision of Course III, nowunder way, is completed, the two volumes of the Course IIIi.ext and the accompanying Teachers Commentary will besimilarly available.

Education of Teachers

Each summer, beginning in 1966, the teachers of experimentalclasses have phrticipated in a six-week program of specialstudy at Teachers College, designed to prepare them toteach an experimental course in the following school year.This program of study wan in two parts. The first was acourse in the mathematical subject areas underlying theexperimental course content, such as: abstract algebra,linear algebra, transformation geometry and probability.These subjects were taught from a modern and unified pointof view, The second part of this program w 5+ course inpedagogical methods of teaching secondary mathematics as aunified branch of knowledge, with special emphasis In thespecific structures and principles to be covered in thefollowing year of experimental teaching.

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The following is a list of the teachers and the schools inwhich they taught the experimental classes (Course III)during the 1968-69 school year:

Elmont, New YorkSchools: Alva T. Stanforth Junior High School

Sewanhaka High SchoolTeachers: Samuel Backer

Alexander ImreEdward KeenanMary Murray

Leonia, New JerseySchool: Leonia High SchoolTeachers: Christine McOoey

David Swaim

New York, New York3Ghool: Hwii er Goilege sign ScnoolTeachers: Douglas Bumby

Ruth CohenRichard Klutch

Teaneet, New JerseySchools: Benjamin Franklin Junior High School

Thomas Jefferson Junior High SchoolTeaneck High School

Teachers: Franklin ArmourAnnabelle CohenOtto KruppMary Renda

Westport, ConnecticutSchools: Bedford Junior High School

Coleytown Junior High SchoolLong Lots Junior High SchoolStaples High School

Teachers: David FlaysRobert KellerJohn PepeDaniel Sullivan

All teachers :Jlowed intense interest and cooperatedsplendidly in acquiring the spirit and content of the pro-posed new curriculum, and in teaching it. As a result ofthis training we now have a core of demonstration teachersand also a body of subject matter that must constituteteacher preparation in the future,

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Teaching Courses

Six junior high schools in the New York Metropolitan areahave taken experimental classes through Course I, II, andIII. Five of these classes covered the text material inits first experimental revision. Another fifteen experi-mental classes have completed these three courses in therevised versions and another fifteen non-experimentalclasses in these schools have completed both Course I and IIin their final revisions, Since the SSMCIS program is atpresent deals ned for those students in roughly the top 15%of their class with respect to mathematical ability, theoriginal selection of students for the twenty experimentalclasses was made by the participating schools with priormathematics achievement and scores on aptitude tests asmain criteria.

Because the teachers of pilot classes were working as a teamin the experimental class, they were often able to helpeach other with difficulties that arose in understanding orteaching the new material. Having had this year of teamteaching experience, the teachers are now prepared to teachCourses I, TI. and III on their oNn.

During the :.1:shonl year, the director and project staffmembers made frequent personal visits to observe theexperimental teaching, Each clans was observed at leastfour times. Visits to these schools included discussionswith the teachers and administrators concerning progressand problems with the experimental course,

The teacher- were further assisted by several full-daymeetings nt Teachers College where teaching problems werereviewed with selected consultants and the project direct-or, At these meetings many teaching difficulties wereresolved and valuable criticisms of the textbook weregathered.

Evaluation of Courses 1.1 II, and III

The six year mathematics program introduces many newconeepts into the secondary school mathematics curriculumand in',egrates both stnnderd and new topics in a globalorganization not characteristic of existing programs,Student achievement in such a program cannot adequately bemeasured using conventional standardized tests, For thisreason, student learning was tested by extramural examina-tions constructed by the project staff,

To guide construction of these and future measurementinstruments, the textbooks were analyzed to produce ataxonomy of cognitive objectives. This taxonomy sided in

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delineating goals of instruction in terms of subject matterand related behaviors. The categories of behavior appearin Table I,

Each year, two examinations were prepared; one for a mid-year evaluation, and one for an evaluation at the end ofthe year. These instruments were used as n measure of theteachability as well as of the learning of the preparedcontent. They also guided the revisions that were sub-sequently made.

Alhough achievement on standardized traditional mathematicstests was not accepted as a measure of the success of theexperimental program, it was of interest to determine atthe start whether or not study in the experimental CourseI affected learning of traditional topics, To accomplishthis objective all students were administered the Sequen-tial Test of Educational Progress - Mathematics, Form 3A -in September 1966 and again in September 1967, The testresults clearly show that students in the project classessuffered no decline in mathematical skills when comparedwith students studying more traditional programs, More-over, the achievement of these students on the projecttests shows that they were leerning to work manynew and powerfUl mathematical tools not r, part of thetraditional mathematics fare of seventh gradora,

Future Activity

The Secondary School Mathematics Curriculum ImprovementStudy recei-ved support from the National Science Poundationin June 1969 to continue designing and experimenting witha unified secondary school program for college cApablestudents in the senior high school. This support hasenabled the project to produce Course IV in the six yearsequence an to begin planning for Course V and VI,

Course iv. whir) had been written in the summer of 1969,was pilot-tentcd in five experimental schools during the190-1970 school year. As a result or this experil.entation,Course IV will be revised during the summer of 1970and will be available by the early Pall (1970). Afterfurther testing, the final version will be availableby September 1971.

A tentstive outline for Course V was developed durinvtwo day conference of the advisory council in Januar:, 1970,lhis outline was expanded during a nine day conferenceof writers and consultants into detailed guides for thewriters or the chapters in course V. This eonrse will bewritten dcrin07. the summer of 1070 and experimented with infive schooln dnring 107u.

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The planning for and writing of Course VI will follow asimilar procedure to that outlined for Course V with class-room testing during the 1971-72 school year.

By the end of the academic year 1973, the SSMCIS will havecompleted its task - to make a reconstruction of thesecondary mathematics curriculum by presenting the subjectAS an integrated body of knowledge reflecting the spiritof --ntemporary mathematics.

Conclusions and Recommendations

After 1 years of classroom experimentations with studentsselected in the upper 15 to 205 or academic ability andtaught by interested classroom teachers with special train-ing in subject matter and pedagogy, a new curriculum inmathematics has been designed for the junior high schoolstudy, grades seven, eight, and nine. This curriculumbreaks down the traditional barriers separating arithmetic,algebra, and geometry, mod unifies the study through thosefundamental concepts underlying all the branches, namelysets, relations, functions, and operations. The resultingcurriculum is like a double helix in which the importantstructures - group, ring, field and vector apace - formone strand, while the other strand consists of the impor-tant realizations: the number systems and the severalgeometries; synethetic, coordinate, vector, and transfor-mation. interwoven with both these strands are the activi-tien and applications including the study of function,conditional sentences, statistics ard probability.

Students who complete this three year program are advancedin knowlede more than one year beyond the present collegepreparatory program. This in accomplished by eliminating agreat deal or traditional content that, today is of little orno value in further study or applicstton or mathematics.The increase in learning is also orolwht shout by the unifi-cation of all the study under the more r,c!neral concepts andstructures of conteporary mathematics.

The teachability has been tested and verified, each courseunlerrpIng, thr.e yearn of thorough examinations and revi-sion into its present form. Concomitantly, five doctoralstudies researching the lecrning and teaching, aspects ofthe program were completed. These studies were ..tarried outby the research assistants associated with the project.They were:

ilicholas A. Branca - ''strategies In 1 earninu V.s'._hernaties1:structures,'' 197(.

James T. Fey - "Patterns of Verbal Communication inMathematics Classes," 1968.

Michael J, Hoban - "Transformation Geometry in the JuniorHigh School: An Evaluation of aCurricular Unit in the 7th Grade,"1970.

Stanley F. Taback - "The Child's Concept of Limit," 1969.

H. Laverne Thomas - "An Analysis of Stages in theAttainment of a Concept of Function,"1969.

A study in one school has shown that the same material canbe learned by students of average ability, if pursued at aslower rate over a longer period of time.

It is strongly recommended that the program developed bySSMCIS be reexamined for adaptation as a curriculum for allstudents in the junior high school. It is also recommendedthat the SSMCIS program serve as a basis for re-examiningthe pre-service mathematics education of prospective juniorhigh school teachers of mathematics. For these teachers,far more attention must be given to abstract and linearalgebra, and to geometry of a modern variety, with lessstress on advanced analysis. Lastly it is recommended thatthe unified approach be extended throughout the senior highschool study to give college preparatory students an ad-vanced knowledge of all aspects of mathematics on enteringcollege and not only that of the calculus.

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TABLE I

TAXONOMY OF OBJECTIVES

Mathematical Objectives

Structures: Arithmetic Geometry Probability Analysisand Algebra

Fundamental Sets Operations Relations Mappings LogicConcepts

Behavioral Objectives

I. Ability to recall definitions, notations, operations,

concepts.

II, Ability to manipulate and calculate efficiently.

III. Ability to interpret symbolic data or processes.

IV. Ability to communicate mathematical ideas.

V. Ability to apply concept to a purely mathematical

situation - -solve problems.

VI. Ability to apply concept to problems in other

situatioresolve word problems.

VII, Ability to transfer learning to a new situation in

mathematics.

VIII. Ability to construct or follow a mathematical arignienit.

Of course not all these categories apply to each subjectmatter topic, but the gcala were checked against subjectmatter,

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APPENDIX A

COURSE I CONTENT

Chapter

1 FINITE NUMBER SYSTEMSJane Anderson's ArithmeticClock Arithmetic(Z12,+) and (W,+)Calendar ArithmeticOpen SentencesNew ClocksRotationsSubtraction in Clock ArithmeticMultiplication in Clock ArithmeticComparison of (W,.) and Clock MultiplicationDivision in Clock ArithmeticInverses in Clock ArithmeticThe Associative and Distributive PropertiesSummary

2 SETS AND OPERATIONSOrdered Pairs of Numbers and AssignmentsWhat is an OperatioA?Computations with OperationsOpen SentencesProperties of OperationsCancellation LawsTwo Operational SystemsWnat is a Group?Summary

3 MATHEMATICAL MAPPINGSAssignments and MappingsMappings of Sets of Whole NumbersMappings of Clock NumbersSequencesComposition of MappingsInverst and Identity MappingSpecial Mappings of W to WSurmary

THE INTEGERS AND ADDITIONIntroductionSome New NumbersThe Integers and OppositesProperties of (Z,+)The Integers and Translations on a LineSubtraction in (Z,+)Subtraction as Addition of OppositesEquations in (Z,+)Cancellation Law

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ChapterOrdering the IntegersAbsolute ValueSummary

5 PROBABILITY AND STATISTICSIntroductionDiscussion of an ExperimentExperiments to be Performed by StudentsThe Probability of an EventA Game of ChanceEqually Probable OutcomesAnother Kind of MappingCounting with TreesPreviewResearch ProblemsStatistical DataPresenting Data in TablesThe Frequency Histogram and the Cumulative

Frequency HistogramSummary

6 MULTIPLICATION OF INTEGERSOperational Systems (MO and (Z,)Multiplication for ZMultiplication of a Positive Integer and a

Negative IntegerThe Product of Two Negative IntegersMultiplication of Integers through DistributivityDilations and Multiplication of IntegersSummary

7 LATTICE POINTS IN THE PLANELattice Points and Ordered PairsConditions on Z x Z and their GraphsIntersection and Unions of Solution SetsAbsolute Value ConditionsLattice PoInt GamesSets of Lattice Points and Mappings of Z into ZLattice Points in spaceTranslation and Z x ZDilations and Z x ZSome Additional Mappings end 2 x ZSummary

8 SETS AND RELATIONSSetsSet Equality and SubsetsUniversal Set, Subsets and Venn DiagramsUnions, Intersections and ComplementsCartesian Product Sets: RelationsProperties of RelationsEquivalence Classes and PartitionsSummary

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Chapter9 TRANSFORMATIONS OF THE PLANE

Knowing How and DoingReflections in a LineLines, Rays and SegmentsPerpendicular LinesRays Having the Same EndpointReflection in a PointTranslationsRotationsSummary

10 SEGMENTS, ANGLES, AND ISOMETRIESIntroductionLines, Rays, Segments?lanes and HalfplanesMeasurements of SegmentsMidpoints and other Points of DivisionUsing Coordinates to Extend IsometriesCoordinates and TranslationsPerpendicular LinesUsing Coordinates for Line and Point ReflectionsWhat is an Angle?Measuring an AngleBoxing the CompassMore about AnglesAngles and Line ReflectionsAngles and Point ReflectionsAngles and TranslationsSum of Measures of the Angles of a TriangleSummary

11 ELEMENTARY NUMBER THEORY(N,+) and (N, .)DivisibilityPrimes and CompositesComplete FactorizationThe Sieve of EratosthenesOn the Number of PrimesEuclid's AlgorithmSummary

12 THE RATIONAL NUMBERSW, Z and Z,Reciprocals of the IntegersExtending Z U Z' to Q(Q,Properties or (Q,.)Division of Rational NumbersAddition of Rational Numbers

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ChapterSubtraction of Rational NumbersOrdering the Rational NumbersDecimal FractionsInfinite Repeating DecimalsDecimal Fractions and Order of the Rational NumbersSummary

13 SOME APPLICATIONS OF THE RATIONAL NUMBERSRational Numbers and DilationsComputation with Decimal FractionsRatio and Proportion.Using ProportionsMeaning of PercentSolving Problems with PercentsPresenting Data in Rectangular, Circle, andBar Graphs

Translations and GroupsApplications of TranslationsSummary

14 ALGORITHMS AND THEIR GRAPHSPlanning a Mathematical ProcessFlow Charts of Branching AlgorithmsInteractive AlgorithmsTruncated Routines and Truncation CriteriaSummary

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APPENDIX P

COURSE II CONTENT

Chapter

1 MATHEMATICAL LANGUAGE AND PROOFIntroductionMathematical StatementsConnectives: And, OrOonditional and Bi-conditional StatementsQuantified StatementsSubstitution Principle for Equality (SPE)InferenceDirect Mathematical Prc.f.Indirect MathemAtical ProofSummary

2 GROUPSDefinition of a GroupA Non-Commutative GroupMore on PermutationsFunctional NotationMore NotationSome Theorems About GroupsIsomorphismSummary

3 AN INTRODUCTION TO AXIOMATIC AFFINE GEOMETRYPreliminary RemarksAxiomsSome Logical Consequences of the AxiomsA Non-Geometric Model of the AxiomsOther Models of the Axioms - Finite and InfiniteEquivalence Classes of Parallel LinesParallel ProjectionVectors - An Intuitive IntroductionSummary

4 FIELDSWhat is a Field?Getting Some Field Theorems PainlesslyTrouble with 0Subtraction and Division in FieldsFractions in FieldsOrder in FieldsHow Many Ordered Fields':Equations and Inequations in (Q,+,,)Solving Quadratic EquationsSummary

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Chapter

5 THE REAL NUMBER SYSTEMThe Equation xe = 2 in (Q,+,.)The Measuring ProcessThe Length of a Line SegmentThree Illustrative CasesThe Real Number SystemSome Properties of the Real Number SystemArithmetic of Irrational NumbersSummary

6 COORDINATE GEOMETRYIntroductionAxiom 4, Uniqueness of Line Coordinate SystemsAxiom 5, Relating Two Coordinate Systems on a LineSegments, Rays, MidpointsAxiom 6, Parallel Projections and Line ProjectionsPlane Coordinate SystemsAn Equation for a LineIntersections of LinesTriangles and QuadrilateralsThe Pythagorean PropertyPlane Rectangular Coordinate SystemsSummary

7 REAL FUNCTIONSMathematical MappingsProperties of Real FUnctionsRepresenting Real FunctionsComposition of Real FUnctionsInverses of Beal Functionsif g) and if g)

iC.gl ond (-1

The Square Root and Cube Root FunctionsSummary

8 DESCRIPTIVE STATISTICSIntroductionExamples of Sets of Data and Their Graphics'.

Presentation

The Symbol , and SummationThe Arithmetic Mean, Its Computation and PropertiesMeasures of DispersionSimplified Computation of the Variance and theStandard Deviation

Summary

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Chapter

9 TRANSFORMATIONS IN THE PLANE: ISOMETRIESWhat is a Transformation?Reflections in a LineTranslationsRotations and Half -TurnsComposing Isometries, Glide ReflectionsThe Three Line Reflection TheoremDirected IsometriesGroups of IsometriesIsometry, Congruence, and SymmetryOther Transformations: Dilations and SimilaritiesSummary

10 LENGTH, AREA, VOLUMEIntroductionMeasures on SetsLengths of Line SegmentsAreas of Rectangular RegionsVolumes of Rectangular SolidsAreas of Triangular RegionsAreas of Parallelograms and Trapezoidal RegionsAreas of other RegionsCircumference of a Circle and TrAreas of Circular RegionsSummary

Appendix A: Mass PointsMass PointsNotations and ProceduresAxioms for Mass PointsA TheoremAnother TheoremUsing a DefinitionMass Points in Space and a TheoremSummary

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APPENDIX C

COURSE III CONTENT

Chapter

1 INTRODUCTION TO MATRICESWhat is a MatrixUsing Matrices to Describe Complex SituationsOperations on MatricesMatrices and Coded MessagesMatriles and TransformationsTransition MatricesSummary

2 LINEAR EQUATIONS AND MATRICESLinear Combinations of EquationsSolving Systems of Linear EquationsSolving Systems of Linear Equations, ContinuedHomogeneous Linear Equatione;,''stems of Linear Equations and MatricesMatrix InversionSummary

3 ALGEBRA MATRICESThe World of MetricosAddition of MatricesMultiplication by a ScalarMultiplication of MatricesMultiplicative Inverses in M,The Rim of 2 x 2 MatricesA Field of 2 x 2 MatricesSummary

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GRAPHS AND iliNCTIONSConditions and GraphsRegions of the Plane and TranslationsFuncttons and ConditionsFunctions and Solution of EquationsOperations on Functions and AsymptotesSumInnry

COMEINATORICSIntroductionCounting Principle and PermutationsT1-1 Power ;et of a SetNumber of Subsets of a Given SizeThe Binomial TheoremMathematical InductionSummary

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Chapter

6 PROBABILITYIntroductionOutcome Set and EventsProbability MeasureUniform Probability MeasureLooking BackLooking AheadSummary

7 POLYNOMIAL AND RATIONAL FUNCTIONSPolynomial FunctionsDegree of a PolynomialAddition of Polynomials (P,+)Multiplication of Polynomial Functions (P,+,)Division of Polynomial FunctionsPolynomial Factors and The Facto TheoremQuadratic Functions and EquationsRational FunctionsOperations with Real Rational FunctionsSummary

8 CIRCULAR FUNCTIONSSensed AnglesStandard PositionCircular PunctionsCircular FunctionsDegree Measure andGraphs of CircularLaw of Cosines andSnmmary

of Anglesof Real NumbersSpecinl AnglesFunctionsLaw of Sines

9 INFORMAL SPACE GEOMETRYSpace Geometry and Plane GeometryPlanes in SpaceParallel Lines and Planes in SpaceDeductive Approach to Geometry in SpacePerpendicularity of Lines and Planes in SpaceCoordlnatfl systems in 3-SpaceSet Descriptions of Planes in Coordinate 3-SpaceSurfaces in SpaceSummary

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