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ED 088 705 SE 017' 485

TITLE School Mathematics Study_Gronp Newsletters[Curriculum Content, Development and ObjectiveNumbers 1-4-6-11-13-24-25-28-33-36-38.

INSTITUTION Stanford Univ., Calif. School Mathematics StudyGroup.

SPONS AGENCY National Science Foundation, Washington, D.C.PUB DATE 72NOTE 288p.

EDRS PRICE MF-$0.75 HC-$13.80DESCRIPTORS Bibliographies; Curriculum; *Curriculum Development;

Elementary School Mathematics; Instruction;*Mathematics Education; *Newsletters; Research;Secondary School Mathematics

IDENTIFIERS *School Mathematics Study Group; SMSG

J.

ABSTRACTThis document is one of three made up of newsletters

from the School Mathematics Study Group's (SMSG) series ofnewsletters written between 1959 and 1972. This set containsnewsletters 1, 4, 6, 11, 24, and 36, covering the history,objectives, organization, and projects undertaken by SMSG;newsletters 13, 25, and 28, discussing the preparation andarticulation of content in SMSG tests; newSletter 33, which discussesmathematics for disadvantaged and low-achieving students; andnewsletter 38, which states objectives and minimum goals formathematics education. Related documents are SE 017.486 and SE 017.487. (JP)

SCHOOLMATHEMATICSSTUDY GROU P

trN

U.S. DEPARTMENT OF HEALTH,EDUCATION &WELFARENATIONAL INSTITUTEOF

EDUCATIONTHIS DOCUMENT HAS BEEN REPRO.

C)- DUCED EXACTLY. AS RECEIVED FROMTHE PERSONOR ORGANIZATION ORIGIN.ATING I T. POINTS OF VIEW OR OPINIONSSTATED DO NOT NECESSARILY REPRE.SENT OFFICIAL NATIONAL INSTITUTEOFEDUCATION POSITION OR POLICY.

Newsletter No. IMarch 1959

TO ERIC' AND ORGANIZATIONS OPERATINGUNDER AGREEMENTS WITH THE NATIONAL IN-STITUTE OF EDUCATION. FURTHER REMO-.DUCTION OUTSIDE THE SYSTEM. REQUIRES PERMISSION OF THE COPYRIGHTOWNER."

"PERMISSION TO REPRODUCE THIS COPY-RIGHTED MATERIAL HAS BEEN GRANTED BY

E. G. Beg leTO ERIC AND ORGANIZATIONS OPERATINGUNOER AGREEMENTS WITH THE NATIONAL IN-STITUTE OF EDUCATION. FURTHER REPRO-DUCTION OUTSIDE THE ERIC SYSTEM RE.QUIRES PERMISSION OF THE COPYRIGHT

OWNER .

To keep the mathematical community informed of its plans

and progress, the School Mathematics Study Group will issue

a newsletter from time to time. Thisfirst.Newsletter, devoted

to the history, organization, present activities, andfuture plans

of the School Mathematics Study Group, is being sent to each

member of the National Council of Teachers of athematic.c.

Those who wish to receive subsequent, issues of the Newsletter

should so indicate, using the form printed on the inside back

cover.

Financial support of the

School Mathematics Study Group

has been provided by the

National Science Foundation

HISTORY

In the spring of 1958, after consulting with the Presi-dents of the National Council of Teachers of Mathe-matics and the Mathematical Association of America,the President of the American Mathematical Society ap-pointed a small conunitt f educators and universitymathematicians to organize a chool Mathematics StudyGroup whose objective would be the improvement ofthe teaching of mathematics in the schools. Professor E.G. Begle was appointed Director of the Study Group,with headquarters at Yale University. In addition, theorganizing committee appointed an Advisory Com-mittee, consisting of college and university mathema-ticians, high school teachers of mathematics, experts ineducation, and representatives of science and technology,to work with the director.

The National Science Foundation, with an initialgrant of $too,000 in the spring of 1958 sand a furthergrant of SI,200,000 later that year, has provided the fi-nancial support for the work of the Study Group.

OBJECTIVESThe world of today demands more mathematical

knowledge on the part of more people than the worldof yesterday and the world of tomorrow will make stillgreater demands. Our society leans more and moreheavily on science and technology. The number of ourcitizens skilled in mathematics must be greatly increased;an understanding of the role of mathematics in our so-ciety is now a prerequisite for intelligent citizenship.Since no one can predict with certainty his future pro-fession, much less foretell which mathematical skills willbe required in the future by a given profession, it is im-portant that mathematics be so taught that students willbe able in later life to learn the new mathematical skillswhich the future will surely demand of many of them.

To achieve this objective in the teaching of schoolmathematics three things are required. First, we need animproved curriculum which will offer students not onlythe basic mathematical skills but also a deeper under-standing of the bask concepts and structure of mathe-matics. Second, mathematics programs must attract andtrain more of those students who are capable of study-ing mathematics with profit. Finally, all help possiblemust be provided for teachers who are preparing them-selves to teach these challenging and interesting courses.

Each project undertaken- by the School MathematicsStudy Group is concerned with one or more of thesethree needs.

ORGANIZATIONBasic policy for the School Mathematics Study Group

is set by the Advisory Committee. This Committeemeets periodically to review the progress of currentprojects and decides on the undertaking of new projects.A list of the members of the Advisory Committee is tobe found on page 5.

An Executive Committee of the AdvisOiy. Com-mittee is available to handle urgent problems betweenmeetings of the Advisory Committee.

Each specific project undertaken by the School Mathe-matics Study Group is supervised by a Panel, drawn inpart from the Advisory_Committee. A list of the mem-bers of the Panels as currently constituted may be foundon pages 6 and 7.

4

ADVISORY COMMITTEE

A. A. Albert, University of ChicagoF. B. Allen, Lyons Township High School, LaGrange, IllinoisE. G. Begle, Yale UniversityLipman Bers, New York UniversityS. S. Cairns, University of IllinoisG. F. Carrier, Harvard UniversityW. L Duren, Jr., University of VirginiaHoward Fehr, Columbia UniversityH. V. Funk, Columbia UniversityA. M. Gleason, Harvard UniversityJ. H. Hlavaty, Dewitt Clinton High School, Ncw YorkP. S. Jones, University of MichiganL Clark Lay, Pasadena City College.K. 0. May, Carleton CollegeJ. R. Mayor, American Association for the Advancement of

ScienceA. E. Meder, Jr., Rutgers UniversityE. E. Moise, University of MichiganP. M. Naghdi, University of CaliforniaRichard Pieters, Phillips Academy, Andover, MassachusettsG. B. Price, University of KansasR. E. K. Rourke, Kent School; Kent, ConnecticutA. W. Tucker, Princeton UniversityHenry Van Engel', University of WisconsinA. D. Wallace, Tulane UniversityE L. Walters, William Penn Scnior High. School, York, Penn-

sylvaniaMark S. Wilcox, Thomas Carr Howe High School, Indian-

apolis, IndianaM. W. Zemansky, The College of the City of New York,

New York

PANEL ON 7TH AND 8TH GRADES

J. A. Brown, State University Teachers College, Oneonta,New York

I..enore John, University of ChicagoB. W. Jones, University of ColoradoP. S. Jones, University of MichiganJ. R. Mayor, American Association for the Advancement of

ScienceP. C. Rosenbloom, University of MinnesotaVeryl Schub, Supervisor of Mathematics, Washington, D.C.

PANEL ON SAMPLE TEXTBOOKS

F. B. Allen, Lyons Township High School, LaGrange, IllinoisEdwin Douglas, The Taft School, Watertown, ConnecticutD. E. Richmond, Williams College, Williamstown, Massachu-

settsC. E. Rickart, Yale UniversityHenry Swain, New Trier Township High School, Winnetka,

IllinoisR. J. Walker, Cornell University

PANEL ON MONCGRAPHS

Lipman Bers, New York University, Ncw YorkH. S. M. Coxeter, University of Toronto, Toronto, Ontario,

CanadaP. R. Halmos, University of ChicagoJ. H. Hlavaty, Dewitt Clinton High School, New YorkN. Jacobsen, Yale UniversityH. 0. Pollak, Bell Telephone LaboratoriesGeorge Polya, Stanford UniversityR. S. Pieters, Phillips Academy, Andover, Massachusetts.H. E. Robbins, Columbia UniversityW. W. Sawyer, Wesleyan UniversityN. E. Steenrod, Princeton UniversityJ. J. Stokcr, New York UniversityLeo Zippin, Queens College, Flushing, New York

PANEL ON TEACHER TRAINING MATERIALS

J. B. Adkins, Phillips Exeter Academy, Exeter, New Hamp-shire

H. F. Fehr, Columbia UniversityJ. L Kelley, University of California -41

'L. C. Lay, Pasadena City CollegeK; 0. May, Carleton CollegeB. E. Meserve, Montclair State Teachers College, Montclair,

New JerseyG: S. Young, University of Michigan(The Director is es officio a member of each panel)

RELATIONSHIPS WITHOTHER ORGANIZATIONS

The School Mathematics Study Group is, of course,not the only organization concerned with the improve-ment of mathematics in our schools. Some of the othersare: the Secondary School Curriculum Committee ofthe National Council of Teachers of Mathematics; theUniversity of Illinois Committee on School Mathe-matics; and the Commission on Mathematics of theCollege Entrance Examination Boird.

The School Mathematics Study Group is not in com-petition with any of these organizations. Instead, webelieve that we can supplement and extend their work.Liaison with them is provided by the simple expedientof including on our Advisory Committee and on ourwriting teams representatives of these organizations.

The Federal Government also is vitally interested inthe improvement of school mathematics, as is evidenced,for example, by the financial support provided by theNational Science Foundation for the School Mathe-matics Study Group. Even more important is the pro-

gram of Summer, Academic Year, and In-Service In-stitutes sponsored by the National Science Foundation,without which any, widespread improvement in theteaching of school mathematics would be virtually im-possible.

PROJECTS UNDER WAYEXPERIMENTAL PROJECTGaanns 7 AND 8

The School Madiematics Study Group believes itparticularly important that greater substance and in-terest be given to the mathematics of grades 7 and 8.One of our major projects now under way is concernedwith the curriculum for these two grades. Unlike thecurriculum for grades 9 through 12, the curriculum forthese two grades has not had the benefit of much, carefulstudy in recent years. Our approach has therefore beenexperimental. About a dozen experimental units havebeen written and are now 'being tried out in approxi-mately one hundred classes scattered over the wholecountry. Some of these experimental units deal withstandard topics treated from a new point otview; otherspresent material that is likely to be novel at this: level.Our general point of view has been to think of grades7 and .8 not as the end of elementary school mathematics,but rather as a foundation for the, work of the seniorhigh schooL We have tried to include in the experi-mental units a better intuitive basis for the algebra-andgeometry courses which should follow.

Preliminary reports from classrooms in which thesechapters are being tried indicate that students find mathe-matics from this new point of view very interesting. .

In this work we have cooperated very closely, sharingboth textbook material and personnel, with a similarproject at the Univeriity of Maryland.

At :.a writing session to be held this coming summer,we plan to produCe a complete. 7th-grade course to-gether with additional supplems-entary units for die 8t1igrade, taking into account this year's classroom experi-ence with our present units.. These new materials willthen be tried out during the academic year ioso-6o andan 8th-grade course will be written during the summer'of 196o.

The foregoing experimental units for grades 7 and '8are now available for inspection. See the ordor form onthe inside back cover.

SAMPLE TEXTBOOKSGRADES 9-12

A second project now under way is aimed at the pro-duction of a series of sample textbooks for grades. 9through 12. The topics discussed in these textbooks willnot differ markedly from those included in the present-day high school courses for these grades. However, theorganization and presentation of these topics will bedifferent and, we hope, improved. The curriculumwhich these textbooks will illustrate will stress both con-cepts and. skills. It is hoped that it will provide a betterunderstanding of the nature of both inductive and de-ductive reasoning and at the same time lead to a betterappreciation of mathematical structure.

We do not believe -that any single curriculum is tobe preferred to the exclusion of all others, and we intendthe curriculum embodied in these textbooks to be nomore than a sample of the kind of curriculum which wehope to see in our schools.

We foresee a variety of ways in which these textbookswill be used. They will, we hope, provide a model anda source of suggestions for the authors of the classroomtextbooks of the future. They will be used in a widevariety of classes to see how new topics in mathematicsand new points of view ars huld,10,.liy students ofdifferent ability levels and by teachers with differentdegrees of preparation. We expect that a major use ofthese textbooks will be in connection with both pre-service and in-service training of teachers, since they willprovide concrete illustrationi of how an increased em-phasis oh basic concepts and on mathematical structurecan be brought into the classroom.

At a writing session held at Yale University duringthe summer of 1958, detailed outlines for these textbookswere prepared, after a careful inspection of the publica-tions and recommendations of other organizations in-.terested in high school mathematics. At a second writingsession to be held at the University of Colorado duringthe summer of 1959, we plan to prepare complete pre-liminary editions of these texts together with accom-panying teachers' guides. These will be used the follow-ing academic year in a variety of classrooms selected bythe School Mathematics Study Group. During the sum-_mer of 196o they will be revised in the light of the ex-

pericnce thus gained and will then be published in non-commercial fashion. They will be kept available for aperiod of about five years, by which time we hope thatthey will have served their purpose.

We plan to make the preliminary editions of thesetexts freely available in the fall of 1959. However, sincethey do present mathematics from a new point of view,classroom use of them is not recommended until the

-- teacher has become thoroughly familiar with their con-tents.

The Panel on Sample Textbooks considered the pos-sibility of publishing the outlines prepared last summer.They decided not to do so in view of their tentativenature. Moreover, it was felt that complete texts wereneeded to contribute effectively to the purposes men-tioned above.

MONOGRAPHS .

A third project is aimed at the production of a seriesof short expository monographs on various mathemati-cal subjects. The primary objectives of such monographsare Ito disseminate good mathematics at the secondaryschool level which will supplement the usual high schoolcurriculum; to awaken interest among gifted students;and to present mathematics as a satisfying, meaningful.human activity. They are not intended as texts, but ratheras supplementary reading material for students, theirteachers, and the general educated lay public.,7

Outstanding mathematicians will write these mono--graphs. In order to be sure that they are understandableand enjoyable by the audience for whom they are in-tended, preliminary versions will be read by high schoolstudents and experienced high school teachers. Theircomments, criticisms, and suggestions will be passed onto the authors to form a basis for revision, if necessrsy.:

The monographs will be published as paper-backs bya commercial publisher. The first few will be availableduring the fall of 1959, and additional ones will appearthe following spring.

TEACHER TRAINING MATERIALS

A fourth project, just getting under way, is devotedto the production of textbooks, study guides, etc., forteachers who wish additional training in mathematics.Particular attention will be paid to the pmduction ofmaterial suitable for use in Summer and In-Service In-stitutes, such as those sponsored by the National ScienceFoundation.

The Panel on Teacher Training. Materials will co-operate with the Committee on the Undergraduate Pro-gram of the. Mathematical Association of America, oneof whose concerns is the pre-service training of mathe-matics teachers.

Two publications in this area are now available fromthe School Mathematics Study Group. The first of theseis a study guide in modern algebra for teachers who wishto improve their professional competence by studyeither individually or in small groups. The second is amonograph "Some Basic Mathematical Concepts"written by Professor R. D. Luce of Harvard Univer-sky. See page r8 for a fuller description of these twopublications.

TESTING OFNEW TEXT MATERIALS

We believe that new text materials should lie thcir-.-oughly tested. We do thii in two Ways. Our experi-mental units for grades 7 and 8, for example, are, beingused in approximately too classrOoms in various parts-of the country'. After finishing each unit, the teacher'fills out a brief questionnaire concerning the ability leVelof the students, the teachability of the unit, and the needfor supplemerc ary material. We plan a innilar treatmeat -for our sample textbooks for grades 9' thitiughea during .the next acadeMic year.

In addition, very careful teats are being carried outfor is-by the Minnesata National Laboratory for thebnprovement of Secondary School Mathematice,. a &-Vision of the Minnesota State Department of EdUcatiOn.Carefully matched experimental and Control clairei and

::teachers use our 7th- and 8th-grade units. Standardachievement tests ire Used. Iii addition; new" tests are,being developed, to measure, for example; attitudestoward and interest in mathematics. '

Furtheiinformation concerning this interesting Labei=ratory will appear in a futUre issue of this Newsletter. :

FUTURE ACTIVITIES OF SMSGThere are a number of other areas in which the School

Mathematics Study Group will carry out studies and,in some cases, undertake projects. Among these are:

I. The use of television, films, and filmstrips both asteaching aids and as teacher training aids. It is hoped thata study of some of the problems in this important anddifficult area will get under way this spring.

11. The sample textbooks for grades 9 through 12 men-tioned above are aimed at the college-capable student.The School Mathematics Study Group has an equal con-cern for the less able student, and work on a programand materials for such students will be undertaken in thenear future. As a first step, arrangements are being madeto try our 9th-grade text with students of intermediateability, in the hope that this material, taken more slowlyand presented more intuitively, will prove both feasibleand useful for such students.

III. Text materials for gifted students are badly needed.Some material of this kind will be included, as optionalsections, in our textbooks. More will be prepared whenthe preliminary editions of the textbooks are completed.

N. A conference on elementary school mathematics,sponsored by the School Mathematics Study Group, washeld in Chicago on February 13 and Li, 1959. This con-ference recommended a comprehensive study of the entire mathematics curriculum from Kindergarten throughgrade 12. Numerous specific suggestions were made con-cerning this study and a committee was requested toformulate detailed plans and to present its reconunenda-dons to the Advisory Committee of the School Mathe-matics Study Group.

V. Plans are being made to enlist the aid of socialscientists, particularly psychologists, in our work. Oneoutcome of the Chicago conference on elementaryschool mathematics was an increased awareness on thepart of all the participants that the contributions of psy-chology to the teaching of mathematics are potentiallyvery great.

THE 95$ .WRITING SESSIONThe outlines of the sample textbooks for grades 9

through 12, and also the experimental units for grades7 and 8 mentioned above, were produced at a writingsession held at Yale University. June 23 to July 18, 1958A list oldie participants in this writing session will befound on pages 16 and 17. It will be observed that theparticipants included both experienced high schoolteachers and distinguished research mathematicians. In-deed, it is a fundamental belief of the School MathematicsStudy Group that :substantial improvement in the schoolmathematics curriculum can result only from cooperation, onan equal basis, of both these groups.

The most important outcome of this, writing sessionwas proof that such cooperation is possible and pro-.ductive. Many of the participants came with a certainamount of scepticism on this point. But this soon van:.ished and was replaced by enthusiasm and a determina-.tion to see the work through to a conclusion. It is thiscognation and enthusiasm that leads,us to believe thatthe Sdiool Mathematics Study. Group can indeed make .

a. substantial contribution to, the improvement of the .teaching of mathematics in' our schools.

PARTICIPANTS IN THE1958 WRITING SESSION

F. B. Allen, Lyons Township High School, LaGrange, Illinois- E. F. Beckenbach, The RAND Corporation, Santa Monica,

CaliforniaEmil Berger, Monroe High School, St. Paul, Minnesota

- R. H. Bing, University of WisconsinJ. A. Brown, State University Teachers College, Oneonta,

New YorkHope Chipman, University High School, Ann Arbor, Michi-

gan- Mary P. Dolciani, Hunter College

Edwin Douglas, The Taft School, Watertown, ConnecticutE. A. Dudley, North Haven High School, North Haven, Con-

necticutFlorence Elder, West Hempstead High School, West Hemp-

stead, New YorkW. E. Ferguson, Newton High School, Newton, Massachu-

settsJoyce D. Fontaine, North Haven High School, North Haven,

ConnecticutEsther 0. Gassett, Clairmore High School, Clairmore, Olda-

homaE. Glenadine Gibb, Iowa State Teachers College, Cedar Falls,

Iowa- R. A. Good, University of Maryland

Martha Hildebrandt, Provisio Township High School, May-_

wood, IllinoisLenore John, University High School, University of ChicagoB.W. Jones, University of Colorado

-M. L. Keedy, University of Maryland- Marguerite Lehr, Bryn Mawr College

Eunice Lewis, Laboratory High School, University of Okla-homa

M. A. Linton, William Penn Charter School, Philadelphia,Pennsylvania

J. R. Mayor, American Association for the Advancement ofScience

K. G. Michaels, North Haven High School, North Haven,Connecticut

-- E. E. Moise, University of MichiganE. P. Northrop, University of Chicago

O. J. Peterson, Kansas State Teachers College, Emporia, KansasR. S. Pieters, Phillips Academy, Andover, MassachusettsH. 0. Pollak, Bell Telephone LaboratoriesG. B. Price, University of KansasPenis 0. Redgrave, Norwich Free Academy, Norwich, Con-

necticutD. E. Richmond, Williams CoaegeC. E. Rickart, Yale UniversityP. C. Rosenbloom, University of MinnesotaHarry Rudemun, Hunter College High School, New York

CityVeryl Schuh, Supervisor of Mathematics, Washington, D.C.Henry SWain, New Trier Township High School, Wituiet.ka,

iinoisA. W. Tucker, Princeton UniversityH. E. Vaughan, University of IllinoisJohn Wagner, University of TexasR. J. Walker, Cornell UniversityA. D. Wallace, Tulane UniversityE. L Walters, William Penn Senior High School, York, Penn-

sylvaniaWilliam Wooton, Verdugo Hills High School, Tujtmga, Cali-

forniaJ. H. Zant, Oklahoma State University

PUBLICATIONS AVAILABLETO ORDER, USE THE FORM ON THE OPPOSITE PAGE AND

SEND TO School Mathematics Study GroupDrawer 2502A Yale StationNew Haven, Connecticut

STUDY GUIDE IN MODERN ALGEBRA

The purpose of this Study Guide is to provide as-sistance to teachers who wish to improve their pro-fessional competence in modem algebra by study eitherindividually or in small groups. It contains a list of basictopics and ideas which should be part of the mathemati-cal' equipment of teachers' of introductory (9th grade)algebra. There is a short bibliography and, for each of

. the basic topics and ideas, detailed references to this.bibliography.

Price: FREE

SOME BASIC MATHEMATICAL CONCEPTS,by R. D. Luc:s

An exposition of elementary set theory, togetherwith illustrations of the use of set concepts in variousparts of mathematics.

Price: $1.00

EXPERIMENTAL UNITS FOR GRADES 7 AND 8

Fourteen experimental units for these grades, togetherwith a teacher's guide for each unit and answers to theproblems.

Price: $2.00

SMSG NEWSLETTER

If you wish to be pLaced on the mailing list for further :issues ofthis Newsletter, so indicate on the opposite page.

8

Please put my name on the SMSG mailing list.

Name

Address

ORDER FORM

copies of STUDY GUIDE INMODERN ALGEBRA

copies of SOME BASIC

MATHEMATICAL CONCEPTSSLoo each $--__

copies EXPERIMENTAL limnFOR GRADES SEVENAND EIGHT

Saxe each $-- :`

Total enclosed

Address to which above order should be sent, if other thinthe one indicated above

Remittance must-- r:empany each order. Please make-checks-payable-to-Ye-University.----

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Newsletter No. 4March 196o

'PERMISSION TO REPRODUCE THIS COPY-.RIGHTED MATERIAL HAS BEEN GRANTED BY

E. _ G. BelleTO EPP:* ORGANIZATIONS OPERATINGUNDER AGREEMENTS WITH THE NATIONAL IN-STITUTE OF EDUCATION,' FURTHER REPRO,DICTION OUTSIDE THE ERIC SYSTEM RE-WIRES PERMISSION OF THE COPYRIGHTOWNER."

This Newsletter may be considered a progress report tothe mathematicians of this country, on the work of the SchoolMathematics Study Group.

The aim of this. Study Groupbetter mathematics in ourschoolsis spelled out in more detail in the following pages,as is also the program being followed in furtherance of thisaim. Here we wish merely to point out that a very sizeablenumber of mathematicians have devoted a great deal of thoughtand effort to developing materials pointing some new directionswhich could lead to substantial improvements in school mathe-

matics.The task, however, is an enormous one and cannot be fully

achieved by any one group. To assure progress, colleges anduniversities will have to make an even greater contributionthan they are now making.

An encouragingly large number of high school teacherswould like to do a better job, and they should be. providedwith all the help they need. More, and improved, summer andin-service institutes are necessary, as well as more summerschool programs designed for teachers and emphasizing sub-ject matter. The pre-service training of teachers should also beimproved to allow us to look forward to a continuing improve-ment the education of our children.

Individual mathematicians can contribute by designing andteaching subject matter courses for teacher training and also byacting as consultants to groups of teachers trying to introduceimproved courses.

If the enthusiasm of all those participating in this StudyGroup can be reflected in the work which must be done, andcan only be done, by our colleges and universities, then wehave a good chance of achieving success in this task which isso important to the welfare of our country.

Financial Support of theSchool Mathematics Study Grouphas been provided by theNational Science Foundation

Copyright xp6o by Yale University.

Permission to reprint is granted provided that proper credit is gwen.

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To achieve this objective in the teaching of schoolmathematics three things are required. First, we need animproved curriculum which will offer students not onlythe basic mathematical skills but also a deeper under-standing of the basic concepts and structure of mathe-matics. Second, mathematics programs must attract andtrain more of those students who are capable of study-ing mathematics with profit. Finally, all help possiblemust be provided for teachers who are preparing them-selves to teach them: challenging and interesting courses.

Each project undertaken by the School MathematicsStudy Group is concerned with one or more of thesethree needs.

ORGANIZATIONBasic Policy for the School Mathematics Study Group

is set by the Advisory Committee which meets periodi-cally to review the progress of current projects and todecide on the undertaking of new projects. Each specificproject is under the supervision of a Panel, drawn inpart from the Advisory Committee. The membershipof the Advisory Committee and of the various Panelsmay be found on pages 12 through 17.

RELATIONSHIPS WITHOTHER ORGANIZATIONS

The School Mathematics Study Group is, of course,not the only organization concerned with the improve-ment of mathematics in our schools. Some of the othersare: the Secondary School Curriculum Committee ofthe National Council of Teachers of Mathematics; theUniversity of Illinois Committee on School Mathe- .

matics; the Commission on Mathematics of the CollegeEntrance Examination Board, and the University ofMaryland Mathematics Project. Liaison with them isprovided by the simple expedient of including on theAdvisory Committee, and in the work of each Panel,representatives of these organizations.

The Federal Government also is vitally interested inthe improvement of school mathematics, as is evidenozd,for example, by the financial support provided by theNational Science Foundation for the School Mathe-matics Study Group. Even more important is the pro-gram of Summer," Academic Year, and In-Service In-stitutes sponsored by the National Science Foundation,without which. any widespread improvement in theteaching of school mathematics would be yirtually im-possible.

PROJECTS. UNDER WAYL MATHEMATICS MR GRADES 7 AND 8

The School Mathematics Study Group believes itparticularly important that greater substance and in-

r. *eft be given to the mathematics of grades 7 and 8.Our general point of view has been to think of grades7 and 8 not as the end of elementary sdioOl mathe-

: matics, but rather as a foundation for the work of thesenior high sChooL The crirricultrin for these grades,

11"shOtild include a sound intuitive basis for the algebraand geometry courses to follOW.

To provide a concrete illustration of this kind of cur;;, textbooks for these two years Were prepared,

at a writing session at the University of Michigan in the:of ,1959. Brief deicriPtions of cheie texts

found on page 19.ACcompanying each of these texts is a comment

fOr the teacher. These commentaries "include not onthe usual materiah (disCnision of teaching problems,SOlutions for the etc.) but ahriCiiicinsiOnianddeeper expositions of the Mathematics.

These texts are now being:used in twelye Experi-`mental Centers; involving about roo teachers anc18,50..,:,students. Preliminary reports 'froth these Centera areMost encouraging and in particUlar teachers and stn=-dents 'alike find Mitheinatics from this. point of viewmuch more interesting...The teachers in these Centers report regularly

tlieir classroom experience!. Revisions of the.texti, tak=into account these reports, will be prepared ;in :the

:.summer of 196o.

These two texts are based on a series of individualunits which were written in the summer of 1958, in awriting session at Yale University, and tried out inclasses during 1958-59. These units are described onpage 19. They have proved useful as supplements to con-ventional texts for these grades and have been madegenerally available for this purpose.

II. MATHEMATICS FOR GRADES 9 THROUGH 12

This project is devoted to the production of a seriesof sample textbooks for grades 9 through 12. For themost part the topics discussed in these textbooks do notdiffer markedly from those included in the present-dayhigh school courses for these grades. However, the or-ganization and presentation of these topics is different.Important mathematical skills and facts are stressed, butequal attention is paid to the basic concepts and mathe-matical structures which give meaning to _these- skillsand provide a logical framework for these facts.

We do not believe that any single curriculum is to bepreferred to the exclusion of all others, and we intendthe curriculum exemplified in these textbooks to be no

more-than-a-sample of the kind of curriculum whichwe hope to see in our schools.

We foresee a variety of ways in which these text-books will be used. They will, we hope, provide amodel and a source of suggestions for the authors ofclassroom textbooks of the future. A major use of thesetexts will be in connection with pre-service and in-service training of teachers, since they will provide con-crete illustrations of how an increased emphasis on basicconcepts and on mathematical structure can be broughtinto the classroom. Finally, they will provide a stop-gaptill such texts become available through the usual chan-nels (in this connection see page so).

Preliminary versions of these texts were prepared at__a_writing_session_held-at-the-University-of-Colorado-in--

the summer of 1959, using detailed outlines which hadbeen prepared at the Yale writing session in the summerof 1958. Brief descriptions of these texts may be foundon pages 20-22.

As with the texts for grades 7 and 8, a commentaryfor the teacher accompanies each text.

These texts are now being used in a total of 37 ex-perimental centers involving approximately 26o teachersand 28,000 students. As in the case of the project de-scribed above, preliminary reports from these centersare most encouraging.

The teachers using these texts report on their class-room experiences and, on the basis of these reports,the texts will be revised during the summer of 196o.

MONOGRAPHS

:...-A third project is aimed, at the production of a seriesof short expository monographs on various mathemati-cal subjects. The primary objectives of such monographsare: to disseminate good mathematics at the secondaryachool level which will supplement the usual high school

- curriculum; to-awakeniaterest among. gifteditudents;and to present mathematics as a satisfying, meaningfulhUman activity. They are not intended as texts, but ratheras supplementary reading material, for students, theirteachers, and the general educated' lay public..

Outstanding mathematicians will waste these mono-graphs. In order to be sure that they are understandableand enjoyable by the audience for whom they are in-

' tended, preliininny versions will be read by highschoolstudents and experienced high school teachers. Theircomment's, criticisms, sand suggestions will be passed onto the authors to form a basis for revision, if necessary.

These monographs will be published as paperbacks bya commercial publisher. First drafts of a half-dozenManuscripts had been piepared at the time of publica-tion of this NEwsurrrEa.

IV. TEACHER TRAINING MATERIALS

Practically all-recommendations-for-improved-secondary school mathematics curricula that have been seri-ously proposed, either by SMSG or by others, involveaspects of mathematics which have not, in the past, beenincluded in the normal. subject matter training ofsecondary school teachers. This project is devoted to

the production of materials specifically for teachers whowish the additional training in mathematics needed toteach Ai iittproved curriculum. Particular attention ispaid to materials suitable for use in summer and in-service institutes, such as those sponsored by the Na-tional Science Foundation.

Two series of publications are under way. The firstis a series of study guides for teachers who wish to im-prove their professional competence by study either in-dividually or in small groups. One of these, in algebra,is already available (see page 18), and others on analysis,geometry, logic, number theory, and probability arein preparation.

The second series consists of brief expositions of..various topics in mathematics designed explicitly forin-service teachers. Two of these are already available(see page 18), and another, for teachers of the SMSGFirst Course in Algebra, will be available at the begin-ning of the summer.

V. MATHEMATICS FOR NON- COLLEGE-BOUND STUDENTS

The sample textbooks mentioned above for grades9 through Is; were written explicitly for college-capable students. This particular project is devoted tothe construction of an improved curriculum for less ablestudents. As a first step it will test the hypothesis thatsuch students can learn the kind of mathematics con-tained in the SMSG 9th- and ioth-grade texts providedthat the ..tilaterial is presented in a less formal fashionand with more concrete illustrations, and provided thatthe students are allowed to proceed at their own pace.Appropriate revisions of these texts will be made in thesummer of 196o in order to carry out this test. A similartest will be conducted with the SMSG texts for grades7 and 8.

8

--VL ELEMENTARY SCHOOL MATHEMATICS

In this project SMSG will undertake a critical studyof the elementary school mathematics curriculum fromthe point of view of: increased emphasis on concepts andMathematical principles; the grade placement of topicsin arithmetic; the introduction of new topics, particu-larly from geometry; and supplementary topics for thebetter students, for example from number theory.

A start on this will be made in the summer of 196oin a writing session which will prepare experimentalmaterials for grades 4-6.

OTHER ACTIVITIES

Text materials for gifted students are badly needed.Some material of this kind is included, as optionalsections in the textbooks mentioned above and othersWill be published separately.fixanotherdirection,-an experiment isnow under waywhich will test the feasibility ofa correspondence coursefor zifted students. This, if successful, will be onemethod for providing for the gifted students located inschools too.small to offer special sections or courses.

Many students develop in school a negative attitudetoward mathematics and hence are lost to science andiedmology. The .SMSG sample textbooks are nowbeing- studied, by a group including both mathernati-cans and social saennsts, to see how they affect atti-tudes toward mathematics.

CLASSROOM USE OFSMSG TEXTSGENERAL POLICY

It was pointed out above (page 6) that the majorpurposes of the SMSG textbooks are, first, to providea concrete illustration of the kind of mathematics cur-riculum we would like to see in our schools, andsecondly to provide materials useful in teacher training.However, numerous inquiries have already reachedSMSG Headquarters concerning the availability of thesetexts for classroom use until similar texts become avail-able through the normal commercial channels. The pur-pose of this section is to answer these inquiries.

When an SMSG text is decided to be satisfactory forclassroom use, it will be published in non-commercialfashion and made generally available. However, it willbe kept available for no more than five years, in thebelief that by the end of this period it will have servedits major purposes.

General procedure with an SMSG text is to restrictclassroom use of the preliminary version to the SMSGExperimental Centers. The text and the teachers' com-mentary are then revised, taking into account this class-room experience. In some cases a text may be heavilydependent on .a preceding text and a second year ofclassroom use in Experimental Centers followed by asecond revision may be necessary.

ACADEMIC YEAR 1960-61

For the academic year 196o-61 SMSG proposes tomake available revised versions of the texts for grades7, 9, 1o, and 12. These revised texts will be furnishedwith much more durable covers than the ones nowused on the preliminary versions. It is believed that withthese covets the classroom life of a text will be at leasttwo- years. -The- price -of -a -new text,-including-the-more-durable cover, will not exceed $2.5o.

It is not known at present if revised versions of the8th- and 11th -grade texts can be made available for thecoming academic year. These two texts are more heavilydependent on preceding SMSG texts than is true forthe others.

k would be of considerable assistance if those whoare contemplating the use of SMSG texts next academic

10

year would so inform SMSG Headquarters, indicatingthe number of copies of each text that might be ordered.

;':.This will make it possible to forward information onexact prices and instructions for ordering as soon as theyare determined.

The following comments may be of some help to.school authorities thinking of adopting SMSG texts.

It should be kept in mind that most secondary schoolteachers, through no fault of their own, were not pro-vided in their pre-service training with the mathe-'nudes which the use of these texts requires. Conse-quently, most teachers will need some help, in the form-of additional training in Mathematics, when teachingthese texts for the first time. However, experience inthe SMSG Experimental Centers suggests that an in

training program, taught by a subject, matteripecialist, either before or during the first year's use ofthe texts, will be quite satisfactory in answer to thisproblem.

Evidence from the SMSG 7th- and 8th-grade Centersindicates that when a teacher teaches an SMSG coursefor the second time, the amount of in-service assistanceneeded and the extra time needed for preparation areboth-dizsdcally-reduced,--and-inmany-cases-disappearentirely.

An important source of the needed additional mathe-matical training is the program of summer and in-service institutes sponsored by the National ScienceFoundation. A nnumber of the NSF summer institutes..in 1960 will concentrate heavily on SMSG courses, andmany others will undoubtedly use SMSG materials assupplements to their regular courses.

However, SMSG cannot itself provide any direct as-sistance to school systems wishing to use SMSG texts.In fact; a basic principle in America is that education islocally controlled and is independent of the FederalGovernment SMSG, which receives all its financial sup-

from the-Fmleral Government-through-the -NationalScience Foundation, therefore wishes to do nothingvihich might be interpreted as an attempt to influencethis local control of education. SMSG will confine itsactivities to the preparation and testing of improvedtexts. The decision to adopt these texts, and the imple-mentation o.f the decision, is entirely up to the localschool systems.

A final comment is that there already is considerableevidence that adoption of SMSG texts is an almost ir-reversible process. Most teachers are very reluctant toreturn to conventional textbooks after once usingSMSG texts.

ADVISORY COMMITTEE

A. A. Albert, University of ChicagoF. B. Allen, Lyons Township High School, LaGrange, IllinoisE. G. Begle, School Mathematics Study Group, Yale UniversityLipman Bers, New York UniversityS. S. Cairns, University of IllinoisG. F. Carrier, Harvard UniversityW. L. Duren, Jr., University of VirginiaHoward Fehr, Columbia UniversityH. V. Funk, Columbia UniversityA. M. Gleason, Harvard UniversityJ. H. Hlavaty, Dewitt Clinton High School, New York

S; Jones, University of MichiganL Clark Lay, Pasadena City C011egeK. 0. May, Carleton CollegeJ. R. Mayor, American Association for the Advancement of

Science--A-.-E-Mederjr. Rutgers University

E. E Moise, University of MichiganP. M. Naghdi, University of CaliforniaRichard Pieters, Phillips Academy, Andover, MassachusettsG. B. Price, University of KansasR. E. K. Rourke, Kent School, Kent, ConnecticutA. W. Tucker, Princeton UniversityHenry Van Engen, University of WisconsinA. D. Wallace, Tulane UniversityE. L. Walters, William Penn Senior High School, York, Penn-

sylvaniaMarie S. :Vila% Thomas Carr Howe High School, Indian-


New York


J. A. Brown, University of Delaware Laboratory SchoolLenore John, University of ChicagoB. W. Jones, University of ColoradoP. S. Jones, University of MichiganJ. R. Mayor, American Association for the Advancement 'of

ScienceP. C. Rosenbloom, University of MinnesotaVeryl Schuh, Supervisor of Mathematics, Washington, D.C.

12


F B. Allen, Lyons Township High School, LaGrange, IllinoisEdwin C. Douglas, The Taft School, Watertown, ConnecticutD. E. Richmond, Williams College, Williamstown, Massachu-

settsC. E. Rickart, Yale UniversityHenry Swain, New Trier Township High School, Winnetka,

IllinoisR. J. Walker, Cornell University.

PANEL ON MONOGRAPHS

Lipman Ben, New York University, New YorkH. S. M. Coxeter, University of Toronto, Toronto, Ontario,

CanadaP. R. Halmos, University of ChicagoJ. H. Hlavaty, Dewitt Clinton High School, New YorkN. Jacobson, Yale UniversityR. S. Pieters, Phillips Academy, Andover, MassachusettsH. 0. Pollak, Bell Telephone LaboratoriesGeorge Polya, Stanford UniversityH. E. Robbins, Columbia UniversityW. W. Sawyer, Wesleyan UniversityN. E. Steenrod, Princeton UniversityJ. J. Stoke:, New York University

Leo- Zippin; Queens-College Flushing;-New -York


J. B. Adkins, Phillips Exeter Academy, Exeter, New Hamp-shire

H. F. Fehr, Columbia UniversityJ. L. Kelley, University of CaliforniaL. C. Lay, Pasadena City CollegeK. 0. May, Carleton CollegeB. E. Meserve, Montclair State Teachers College, Upper Mont-

clair, New JerseyG. S. Young, Tulane University

PANEL ON NON-COLLEGE-BOUND STUDENTS

V.41:1424-1,-Haag, Yan Witsliall CollegeMildred Keifer, Cincinnati Board of EducationOscar Schaaf, South Eugene High School, Eugene, OregonM. A. Sobel, Montclair State College, Upper Montclair, New

JerseyMarie Wilcox, Thomas Cur Howe High School, Indianapolis,

IndianaA. B. Willcox, Amherst College

3

PANEL ON ELEMENTARY MATHEMATICS

E. Glenadine Gibb, Iowa State Teachers College, Cedar Falls,Iowa

W. T. Guy, University of TexasS. B. Jackson, University of MarylandIrene Sauble, Detroit Public SchoolsM. H. Stone, University of ChicagoJ. F. Weaver, Boston University(The Director is ex officio a member of each Panel)

PARTICIPANTS IN THEWRITING SESSIONS

Below are listed the names of all those who partici-pated

-

in one or more of the three SMSG writing sessions.'It will be observed that the participants induded bothexperienced high school teachers and distinguished,re-search mathematicians. Indeed, it is a fundamental beliefof the School Mathematics Study Group that substantial int- .

provement in the school mathematics curriculum can resultonly from cooperation, on an equal basis, of both these groups.

The mere existence of the sample textbooks for grades7-sa-is-ample-proof-that-stIch-coOperaon-is-not-only

possible, but also productive.

H. W. Alexander, Eulham CollegeF. B. Allen, Lyons Township High School, LaGrange, IllinoisR. D. Anderson, Louisiana State UniversityB. H. Arnold, Oregon State CollegeAlexander Beck, Olney High School, Philadelphia, Pennsyl-

vaniaE. F. Beckenbach, University of California at Los AngelesE. G. Begle, School Mathematics Study Group, Yale UniversityPaul Berg, Stanford UniversityEmil Berger, Monroe High School, St. Paul, MinnesntaArthur Bernhart, University of OklahomaR. H. Bing, University of WisconsinA. A. Blank, New York Uniiersity

----ShirlerBosellr-Franklin-High-School,-SeattlerWashington--7-7J. A. Brown, University of DelawareK. E. Brown, Department of Health, Education and Welfare,'::'

Washington, D.C.J. M. Calloway, Carleton CollegeHope Chipman, University High School, Ann Arbor, MichiganR. R. Christian, University of British ColumbiaR. J. Clark, St. Paul's School, COncord, New Hampshire,Mildred Cole, East Aurora Junior High School, Aurora, MinoisB. H. Colvin, Boeing Scientific Research Laboratories

14

J. A. Cooley, University of TennesseeP. H. Daus, University of California at Los AngelesIL B. Davis, Syracuse UniversityR. A. Dean, California Institute of TechnologyCharles DePrima, California Institute of TechnologyMary Dolciani, Hunter CollegeEdwin C. Douglas, The Taft School, Watertown, ConnecticutFloyd Downs, East High School, Denver, ColoradoE. A. Dudley, North Haven High School, North Haven, Con-

necticutLincoln Durst, The Rice InstituteFlorence Elder, West Hempstead High School, West Hemp-

stead, New YorkMarion Epstein, Educational Testing Service, Princeton, New

JerseyW. E. Ferguson, Newton High School, Newtonville, Massachu-

settsN. J. Fine, University of PennsylvaniaJoyce D. Fontaine, North Haven High School, North Haven,

ConnecticutF. L. Friedmar, Massachusetts Institute of TechnologyHelen Garnets, University of MarylandEsther. Gassett, Claremore High School, Claremore, OklahomaH. M. Gehman, University of BuffaloRonald Genise, Brentwood Junior High School, Brentwood,

New-YorkR. K. Getoor, University of WashingtonE. Glenadine Gibb, Iowa State Teachers College, Cedar Falls,

IowaR. A. Good, University of MarylandV. H. Haag, Franklin and Marshall CollegeAlice Hach, Racine Public Schools, Racine, WisconsinR. It Hartman, Edina- Momingside Senior High School, Edina,

MinnesotaM. H. Heins, University of IllinoisEdwin Hewitt, University of WashingtonMartha Hildebrandt, ProvisO Township High School, May-

wood, IllinoisS. B. Jackson, University of MarylandLenore John, University High School, University of ChicagoB. W. Jones, University of ColoradoP. S. Jones, University of MichiganR. C. Jurgensen, Culver Military Academy, Culver, IndianaH. T. Karnes, Louisiana State UniversityM. L Keedy, University of MarylandMildred Keiffer, Cincinnati Public Schools, Cincinnati, Ohio

Joseph Lehner, Michigan State UniversityMarguerite Lehr, Bryn Mawr College,,Kenneth. Leiscuing, University of Michigan

15

Eunice Lewis, Laboratory High School, University of OklahomaM. A. Linton, William Penn Charter School, Philadelphia,

PennsylvaniaA. E. Livingston, University of WashingtonL H. Loomis, Harvard UniversityNick Lovdjief& Anthony Junior High School, Minneapolis,

MinnesotaR. V. Lynch, Phillips Exeter Academy, Exeter, New Hamp,

shireHelen Marston, Douglass CollegeJ. R. Mayor, American Association for the Advancement of

ScienceW. K. McNabb, Hockaday School, Dallas, TexasS. S. Meyers, Educational Testing Service, Princeton, New

JerseyK. G. Miclueb, North Haven High School, North Haven,

ConnecticutMuriel Mills, Hill Junior High School, Denver, ColoradoE. E. Moise, University of MichiganE. P. Northrop, University of ChicagoO. J. Peterson, Kansas State Teachers College, Emporia, KansasB. J. Pettis, University of North CarolinaR. S. Pieters, Phillips Academy, Andover, MassachusettsC. F. Pinzka, University of CincinnatiH. 0. Poll* Bell Telephone Laboratories

--Walter Prenowitz,-Brooklyn CollegeG. B. Price, University of KansasA. L. Putnam, University of ChicagoPersia 0. Redgrave, Norwich Free Academy, Norwich, Con-

necticutMina Rees, Hunter CollegeD. E. Richmond, Williams CollegeC. E. Rickart, Yale UniversityP. C. Rosenbloom, University of MinnesotaElizabeth Roudebush, Seattle Public- Schools, Seattle, Wash-

ingtonHarry Ruderman, Hunter College High School, New York

CityGeorge Schaefer, Alexis I. Dupont High. School, Wilmington,

DelawareVsnd Schuh, Washington Public Schools, Washington, D.C.J. T. Schwartz, New York UniversityA. L. Shields, University of Michigan0. E. Stanaitis, St. Olaf CollegeRobert Starkey, Cubberley High Schools, Palo Alto, CaliforniaPhillip Stucky, Roosevelt High School, Seattle, WashingtonHenry Swain, New Trier Township High School, Winnetka,

Illinois

i6

Henry Syer, Kent School, Kent, ConnecticutG. B. Thomas, Massachusetts Institute of TechnologyA. W. Tucker, Princeton UniversityH. E. Vaughan, University of IllinoisJohn Wagner, University of TexasRay Walch, Westport Public Schools, Westport, ConnecticutR. J. Walker, Cornell UniversityA. D. Wallace, Tulane UniversityE. L Walters, William Penn Senior High School, York, Penn-

sylvaniaF. C. Watson, East High School, Rochester, Neiv YorkWarren White, North High School, Sheboygan, WisconsinD. V. Widder, Harvard UniversityA. B. Willcox, Amherst CollegeWilliam Wooton, Pierce Junior College, Woodland Hills, Cali-

forniaJ. H. Zant, Oklahoma State University

17

PUBLICATIONS AVAILABLE.TO ORDER, USE THE FORM ON THE INSIDE BACK COVER AND

SEND TO School Mathematics Study GroupBox 2029 Yale StationNew Haven, Connecticut


The purpose of this Study Guide is to provide as-sistance to teachers who wish to improve their pro-fessional competence in modern algebra by study eitherindividually or in small groups. It contains a list of basictopics and ideas which should be part of the mathemati-cal equipment of teachers of introductory (9th grade)algebra. There is a short bibliography and, for each ofthe basic topics and ideas, detailed references to thisbibliography.

Price: FREE--

STUDIES IN MATHEMATICS

VOLUME I. SOME BASIC MATHEMATICALCONCEPTS

by R. D. Luca

An exposition of elementary set theory, together withillustrations of the use of set concepts in various parts ofmathematics.

Price: Si.00

VOLUME 2. EUCLIDEAN GEOMETRY BASED ONRULER AND PROTRACTOR AXIOMS (revisededition)

by C. W. Curtis, P. H. DAUS, and R. J. WALKER

A study, written for high school teachers, of the ap-proach -to -Euclidean-Geometry- used -in- the-SMSG--tenth-grade textbook.

Price: $1.00

NOTE: This revised edition differs from the preliminaryedition published in the spring of 1959 in that Chapter6 of the preliminary edition was completely rewritten'and a new Chapter 8 was added. Owners of the pre- = ,liminary edition may obtain, at no cost, an addendumcontaining this new material.

JUNIOR HIGH SCHOOL MATHEMATICSUNIT'S

These are the materials for grades 7 and 8 prepared bySMSG in the summer of 1958 and tried in ExperimentalCenters in the academic year 1958-59. They have beengrouped into three volumes. Volume I, "Number Sys-tems" discusses numeration, natural numbers, factoringand primes, tests for divisibility, rational numbers, andabstract mathematical systems. Volume II, "Geometry,"discusses non-metric geometry, informal geometry, andmeasurement. Volume III, "Applications," contains a,unit, entitled "What is. Mathematics and Why YouNeed to Know It," and other units on the lever, statis-tics, and chance,

Each volume is accompanied by a Commentary forTeachers. In these are included digests of the reportsfrom the teachers who taught these Units' in theirclasses. These Units are now available for classroom use.The prices for these are: VOLUME I, $ .75, VowiniS..so, VOLUME III, $ .25.

The price of a Commentary for Teachers is the sameas the price of the volume which it accompanies.

MATHEMATICS FOR JUNIOR HIGH SCHOOL

VOLUMES I AND IL

Emphasized in these texts are the following importantideas of junior high school mathematics: structure ofarithmeik from an algebraic viewpoint; the real numbersystem as a progressing development; metric and non-metric relations in geometry. These ideas are con-stantly associated with their applications. Time is givento the topics of measurement and elementary statistics.Careful attention is paid to the appreciation of abstractconcepts, the role of definition, development of precisevocabulary and thought, experimentation, and proof.

Materials are chosen with. the intent to capture the fas-cinating features of mathematics, creation and discovery,rather than just utility alone. The texts provide: adequatepreparation for the remaining work in the SMSG se-quence.

MATHEMATICS FOR HIGH SCHOOLFIRST COURSE IN ALGEBRA

This text emphasizes the structure of algebra. Thestudy of algebra is based on the exploration of the be-havior of numbers. Careful attention is paid to the lan-guage of the subject. While most of the material whichtraditionally makes up a first course in algebra is woveninto the text, this material appears in an entirely differentlight since it is organized and motivated by fundamentalstructure considerations. This preliminary edition doesnot presuppose' a study of the SMSG Junior HighSchool mathematics program.

Price: $4.25

GEOMETRY

This text is designed for the one-year introductorycourse in geometry which is usually taught in the tenthgrade. The book is devoted mainly to plane geometry,with some chapters on solid geometry, and an intro-duction to analytic geometry. Solid geometry i3 introduced very early in the text and is used as a foil to de-velop the students' space perception. There are a con-siderable number of formal proofs in solid geometry,in some cases integrated with the plane geometry andin other cases in special chapters. The book is writtenin the belief that the traditional content of Euclideangeometry amply deserves the prominent place which itnow holds in the high school curriculum. Changes weremade only when the need for them areared to be com-pelling. The basic scheme in the postulates is that ofGeorge D. Birkhoff: It is assumed that the students arefamiliar with the number line, and numbers are usedfreely for measuring both distances and angles. Geometryis connected with algebra at every reasonable oppor-tunity. Statements of definitions and theorems are exactwithout making exactitude a substitute for intuitive in-sight. The text is written in the belief that intuition andlogic should move forward hand in hand.

Price: $3.75

20

INIERMEDIATE MATHEMATICS

This text is predicated on the philosophy that a stu-dent's interest in mathematics grows as he becomesaware of difficulties, overcomes them, and, on the basisof the mastery thus gained, becomes aware of greaterdifficulties. It is written with the feeling that it is in theeleventh grade where able students reach higher groundin a 'mathematical sense. Careful attention has beentaken to give the student some insight into the natureof mathematical thought as well as to prepare him toperform certain manipulations with facility. Includedin the text are chapters devoted to trigonometry,vectors, logarithms, mathematical induction, and com-plex numbers written with a much higher degree ofsophistication than usually appears at this level. Thetext continues in the vein of the ninth-grade material,in that structure is emphasized at all times. For the bene-fit of students who have not previously studied algebrafrom the point of view of structure, an introductorychapter is provided which reviews. elementary algebrafrom this point of view.

Price: $4.75

ELEMENTARY FUNCTIONS

Again, as in the previous texts, there may be a con-siderable amount of over-lap with previous SMSGbooks. This will continue to be so as long as studentsbegin the sequence at varying grade levels. The text isdesigned for use in the first half of the twelfth grade butby judicious use of supplementary material it couldserve as a basis for a longer course. The central themeis a study of functions: polynomial, exponential, loga-rithmic, and trigonometric, with emphasis on practicalapplications wherever possible. The introduction of asimple but geometrically meaningful method for han-dling areas, tangents, and maximum-minimum prob-lems furnishes the student with a good intuitive back-ground for a later course in calculus.

Price: $2.75

2I

INTRODUCTION TO MATRIX ALGEBRA

This text is designed for the last half of the 12th grade.It is devoted to a study of matrices, including applica-tions to solutions of systems of linear equations and togeometry. At the same time careful attention is devotedto algebraic structure, but not to the point where abarren presentation results. Mathematics is introducedwhich is new to the student and the structure is de-veloped as the text proceeds. It is the intent of the textto put the student close to the frontiers of mathematicsand to provide striking examples of patterns that arisein the most varied circumstances. It provides an effectivr,language and some dynamic concepts that will proveuseful in many college courses. A special set of "ResearchExercises" is appended in the hope that some studentsmay be introduced to real mathematical research. Thecommentary for teachers will be announced in the nextNEWSLETTER.

Price: $1.00

If you are not now on our mailing list but wish toreceive further issues of this NEWSLETTER, please request,by' means of a post card, that your name be added to themailing list.

22

ORDER FORM

All SMSG publications are priced at cost. No re-ductions can be offered for bulk orders.


copies FREE


Volume I. copies at $Loo $Volume 2. copies at $I.00 $

JUNIOR HIGH SCHOOL MATHEMATICS UNITS

Complete set copies at $z.75* $


Volume I.Volume II.

copies at $3.5Crt $copies at $2.00t $

MATHEMATICS FOR HIGH SCHOOL

at $425t $First Course in Algebra copiesGeometry copies at $3.75f $Intermediate

Mathematics copies at $475t $Elementary Functions copies at $2.75t $Introduction to

Matrix Algebra copies at $I.00 $

Total Enclosed $

For prices of individual volumes, see p. 19.tThis is the price for both the student and teacher volumes.

These are not sold separately.

Address to which above order should be sent.

Remittance must accompany each order. Please makechecks payable to Yale University, and mail to

SCHOOL MATHEMATICS STUDY GROUP

Box 2029 Yale Station

New Haven, Connecticut

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SCHOOLMATHEMATICSSTUDY GROU P


E. G. BegleTO ERIC AND ORGANIZArONS OPERATINGUNDER AGREEMENTS WITH THE NATIONAL IN-STITUTE OF EDUCATION. FURTHER REPRO-DUCTION OUTSIDE THE ERIC SYSTEM RE-QUIRES PERMISSION OF THE COPYRIGHTOWNER."

Newsletter No. 6March 1961

DO NOT USE FOR ORDERINGSee Current Newsletters,No. i c and later, foravailable publications.

This Newsletter may be considered a progress report tothe mathematicians of this country, on the work of the SchoolMathematics Study Group.

The aim of this Study Groupbetter mathematics in ourschoolsis spelled ow in more detail in the following pages,as is also the program being followed in ftirtherance of thisaim. Here we wish merely to point out that a very sizeablenumber of mathematicians have devoted a great deal of thoughtand effort to developing materials pointing some new directionsWhich could lead to substantial improvements in school mathe-matics.

The task, however, is an enormous one and cannot be fillyachieved by any one group. To assure progress, olives anduniversities will have to make an even greater contributionthan they are now making.

An encouragingly large number of high school teacherswould like to do a better job, and they should be providedwith all the help they need. More, and improved, summer andin-service institutes are necessary, as well as more summerschool programs designed for teachers and emphasizing sub-ject matter. The pre-service training of teachers should also beimproved to allow us to look forward to a continuing improve-ment in the education of our children.

Individual mathematicians can contribute by designing andteaching subject matter courses for teacher training and also byacting as consultants to groups of teachers trying to introduceimproved courses.

If the enthusiasm of all those participating in this StudyGroup can be reflected in the work which must be done, andcan only be done, by our colleges and universities, then wehave a good chance of achieving success in this task which isso important to the welfare of our country.

Financial Support of theSchool Mathematics Study Grouphas been provided by theNational Science Foundation

Copyright 1961 by Yale University.

Permission to reprint is granted provided that proper credit is given.

Printed in the U.S.A.

HISTORY

In the spring of 1958, after consulting with the Presi-dents of the National Council of Teachers of Mathe-matics and the Mathematical Association of America,the President of the American Mathematical Society ap-pointed a small committee of educators and universitymathematicians to organize a School Mathematics StudyGroup whose objective would be the improvement ofthe teaching of mathematics in the schools. Professor E.G. Begle was appointed Director of the Study Group,with headquarters at Yale University. In addition, theorganizing committee appointed an Adirisory Com-mince, consisting of college and university mathema-ticians, high school teachers of mathematics, experts ineducation, and representatives of science and technology,to work with the director.

The National Science Foundation, through a series ofgrants, has provided very substantial financial supportfor the work of the Study Group.

OBJECTIVES

The world of today demands more mathematicalknowledge on the part of more people than the worldof yesterday and the world of tomorrow will makestill greater demands. Our society leans more and moreheavily on science and technology. The number of ourcitizens skilled in mathematics must be greatly increased;and understanding of the role of mathematics in oursociety is now a prerequisite for intelligent citizenship.Since no one can predict with certainty, his future pro-fession, much less foretell which mathematical skills will

be required in the future by a given profession, it is im-portant that mathematics be so taught that students willbe able in later life to learn the new mathematical skillswhich the future will surely demand of many of them.

To achieve this objective in the teaching of schoolmathematics three things are required. First, we needan improved curriculum which will offer students notonly the basic mathematical skills but also a deeper un-

3

derstanding of the basic concepts and structure of mathe-matics. Second, mathematics programs must attract andtrain more of those students who are capable of studyingmathematics with profit. Finally, all help possible mustbe provided for teachers who are preparing themselvesto teach these challenging and interesting courses.

Each project undertaken by the School MathematicsStudy Group is concerned with one or more of thesethree needs.

ORGANIZATION

Basic Policy for the School Mathematics Study Groupis set by the Advisory Committee which meets periodi-cally to review the progress of current projects and todecide on the undertaking of new projects. Each specificproject is under the supervision of a Panel, drawn in partfrom the Advisory Committee. The membership of theAdvisory Committee and of the various Panels may befound on pages 13 through 14.

PROJECTSTSI. MATHEMATICS FOR GRADES 7 AND 8

The School Mathematics Study Group believes itparticularly important that greater substance and in-terest be given to the mathematics of grades 7 and 8.Our general point of view has been to think of thesegrades not just as the end of elementary school mathe-matics but also as a foundation for the work of thesenior high school. The curriculum for these gradesshould include a sound intuitive basis for the algebraand geometry courses to follow.

To provide a concrete illustration of this kind of cur-riculum, textbooks for these two years have been pre-pared. Descriptions of these texts may be found onpage zo.

Accompanying each of these texts is a commentaryfor the teacher. These commentaries include not onlythe usual materials (discussion of teaching problems,solutions for the exercises, etc.) but also discussions anddeeper expositions of the mathematics.

These texts evolved from a series of experimentaljunior high school units which were prepared at the

4

first SMSG writing session at Yale University in thesununer of 1958. In the following summer a prelimi-nary version of a 7th grade text and a collection of 8thgrade units, including much of the material in these ex-perimental units was prepared. In the summer of 196o.a revised version of the 7th grade text and a preliminaryversion of an 8th grade text were prepared. A revisionof the 8th.grade text will be carried out in the summerof 1961.

During. the intervening academic years, these ma-terials were used in a number of experimental centers,involving approximately too teachers and 8,000 studentseach year. Feedback from the teachers was used exten-sively each summer in the revision of these materials.

II. MATHEMATICS FOR GRADES 9 THROUGH 12

This project is devoted to the production of a seriesof sample textbooks for grades g through 12. For themost part the topics discussed in these textbooks do notdiffer markedly from those included in the present-clayhigh school courses for these grades. However, the or-ganization and presentation of these topics is different.Important mathematical skills and facts are stressed, butequal attention is paid to the basic concepts and mathe-matical structures which give meaning to these skills andprovide a logical framework for these facts.

Preliminary versions of these texts were prepared ata writing session held at the University of Colorado inthe summer of 1959, using detailed outlines which hadbeen prepared at the Yale writing session in the summerof 1958.

As with the texts for grades 7 and 8, a commentary. for the teacher accompanies each text.

These texts were used in a total of 37 experimentalcenters involving approximately 26o teachers and i8,000students. As in the case .of the project described above,the teachers using these texts reported on their classroomexperiences and, on the basis of these reports, thetextswere revised during the summer of 1960 at a writingsession at Stanford University. Brief description of thesetexts may be found on pages 22-27.

During the summer of 1961 a preliminary version ofanother geometry text will be prepared. This text willemphasize analytic geometry, and will be tried out in anumber of experimental centers during 1961-62.

5

MONOGRAPHS

A third project is aimed at the production of a seriesof short expository monographs on various mathe-matical subjects. The primary objectives of such mono-graphs are: to disseminate good mathematics at thesecondary school level which will supplement the usualhigh school curriculum; to awaken interest amonggifted students; and to present mathematics as a satis-fying, meaningful human activity. They are not in-tended as texts, but rather as supplementary readingmaterial for students, their teachers, and the generaleducated lay public.

Outstanding mathematicians will write these mono-graphs. In order to be sure that they are understandableand enjoyable by the audience for whom they are in-tended, preliminary versions will be read by high schoolstudents and experienced high school teachers. Theircomments, criticisms, and suggestions will be passed onto the authors to form a basis for revision, if necessary.

These monographs will be published as paperbacks bya commercial publisher. The first six will be ready latein the spring of 1961, at which time a special issue ofthis NEWSLETTER will include information on prices andinstructions for ordering them.

The first six monographs are:Numbers: Rational and Irrational, by IVAN NIVEN.What is Calculus'About?, by W. W. SAWYER.An IlitrathICH011 to Inequalities, by EDWIN BECRENBACH

and RICHARD BELLMAN.

Geometric Inequalities, by NICHOLAS D. KAZARINOFF.

The Contest Problem Book. Problems from the Annual High-School Contests of the Mathematical Association of Ameri-

Ca, compiled by CHAkLES T. SALILIND.The Lore of Large Numbers, by PHILIP J. DAVIS.

IV. TEACHER TRAINING MATERIALS

Practically all recommendations for'improved second-ary school mathematics curricula that have been seri-ously proposed, either by SMSG or by others, involveaspects of mathematics which have not, in the past, beenincluded in the normal subject matter taining of second-ary school teachers. This project is devoted to theproduction of materials specifically for teachers whowish the additional training in mathematics needed to

6

11

teach an improved curriculum. Particular attention ispaid to materials suitable for Ilse in summer and in-service institutes, such as those sponsored by the Na-tional Science Foundation.

Two series of publications are under way. The first isa series of study guides for teachers who wish to im-prove their professional competence by study eitherindividually or in small groups. A study guide in al-gebra is currently being revised and others on analysis,geometry, logic, number theory, and probability are inpreparation.

The second series consists of expositions of varioustopics in mathematics designed explicitly for in-serviceteachers. Five of these are already available (see page r9),and others will be available in the near future.

V. MATHEMATICS FOR LESS ABLESTUDENTS

The sample textbooks mentioned previously forgrades 9 through 12 were written explicitly for col-lege-capable students. This particular project is devotedto the construction of an improved curriculum forslower students. This work is under the guidance of thePanel on Underdeveloped Mathematical Talent. As afirst step it will test the hypothesis that such students canlearn the kind of mathematics contained in the SMSG9th and loth grade texts provided that the material ispresented in a less formal fashion and with more con-crete illustrations, and provided that the students areallowed to proceed at their own pace. Appropriate re-visions of these texts were made in the summer of 1960in order to carry out this test. A similar test is beingconducted with the SMSG texts for grades 7 and 8.

Pending definite information on the feasibility ofthese materials the books are not available for generaldistribution at this time.

VI. ELEMENTARY SCHOOL MATHEMATICS

In this project SMSG will undertake a critical studyof the elementary school mathematics curriculum fromthe point of view of increased emphasis on conceptsand mathematical principles; the grade placement oftopics in arithmetic; the introduction of new topics,particularly from geometry; and supplementary topics

7

for the better students, for example from numbertheory.

A start on this was made in the summer of 196o in awriting session at-Stanford University. A completecourse for grade 4 and selrcted units for grades 5 and 6were prepared. A comme !-..'ary for the teacher accom-panies each unit.

These units are now being used in a total of 27 ex-perimental centers involving approximately 12,000 stu-dents with 1 so fourth grade, Ito fifth grade, and nosixth grade teachers.

These teachers report their classroom experiences and,on the basis of these reports, the units will be revisedduring the summer of 1961. The work for grades 5 and6 will be completed, and during the academic year;1961-62, a complete course of study for mathematicsgrades 4, 5, and 6 will be classroom tested.

. A brief description of the current preliminary elemen-tary school mathematics will be found on pages 30-31.

OTHER ACTIVITIESText materials for gifted students are badly needed.

Some material of this kind is included, as optional sec-tions in the textbooks mentioned above and others havebeen published separately. See page 28. Detailed plansare being considered for a considerable expansion ofthe program of publication of brief supplementary ma-terials for students of varied ability levels.

In another direction, an experiment is now under waywhich will test the feasibility of a correspondence coursefor gifted students. This, if successful, will be onemethod for providing for the gifted students located inschools too small to offer special sections or courses.

Many students develop in school a negative attitudetoward mathematics and hence are lost to science. andtechnology. The SMSG sample 'textbooks are nowbeing studied, by a group including both mathematiciansand social scientists, to see how they affect attitudestoward mathematics.

The Educational Testing Service is currently con-ducting a study of the performance of students usingSMSG texts as compared to those using conventionaltexts.

A careful study of teaching machines, programmedlearning, etc., will be carried out with special referenceto the SMSG text materials. .

8

GENERAL INFORMATIONON USES OF SMSG TEXTS .

There are a number of uses to which the SMSG textscan be put. They will, it is believed, provide a modeland a source of suggestions for the authors of futureclassroom textbooks. A second major use of these textswill be in connection with the pre-service and in-servicetraining of teachers, since they provide concrete ills-. .

trations of how an increased emphasis on bask conceptsand on mathematical structure can be brought into theclassroom. Finally, they will provide a stop-gap untilsimilar texts become available through the usual chan-nels.

A number of inquiries have already been receivedconcerning the possibility of incorporating. SMSG textmaterials in contemplated commercial. textbooks. Theadvisory comniittee of SMSG has approved the follow-ing statement of policy; on this matter.

SMSG has prepared a series of sample texts designed to

illustrate, in a concrete fashion, the kind of curriculum whichthe Members of SMSG believe should be taught iu the schools

of this country. It is not intended that the SMSG texts be con-

sidered as the only suitable ones for such a curriculum. Indeed__

a variety of texts ofthe same general nature is not only pos-

sible but also highly desirable. It is expected that these willbecome available through the usual commercial' channels inthe .near future.

A major purpose of an SMSG text is to serve as a modeland as e.course of suggestions and ideas for the authors of this

variety of texts. Textbook writers should feel free to useSMSG texts in -this way and to adapt, expand, and improvethem for their .own purposes. SMSG would appreciate beinggiven proper credit in such cases, but at the same time itshould be made clear that no endorsement by SMSG is ian-

plied.

Pending the availability of commercial texts along thegeneral lines advocated by SMSG, each SMSG text willbe kept ,available as long as it is needed. There 'will be anannual review, and when it is determined that a text is nolonger needed, it will no longer be kept available. Suitablewarning of such a decision will be given.

9

The SMSG texts are copyrighted and may not be repro-duced without permission. Normally, permission will begranted for the reproduction of selections from the SMSGtexts for purposes consistent with those of SMSG.

As to classroom use of SMSG texts, the general pro-cedure has been to restrict such use of preliminary ver-sions to the SMSG Experimental Centers. The text andteacher's commentary are revised, taking into accountthis classroom experience, as many times as seems neces-sary.

During the course of development of an SMSG text,the preliminary versions are usually made available forinformation purposes and are sold at cost.

When it is decided that a particular SMSG text is ina satisfactory state from the point of view of the threepurposes listed above, the text then is made availablein a non-commercial fashion.

At the present time final revisions of the followingtexts have been completed: Mathematics for JuniorHigh School, Vol. I, First Course in Algebra, Geometry,Intermediate Mathematics, Elementary Functions, andIntroduction to Matrix Algebra. It is expected that afinal revision of Mathematics for Junior High School,Vol. II will be completed in the summer of 1961.

Descriptions of these texts will be found on pages 20through 27.

I0

AVAILABILITY OF SMSGTEXTS FOR CLASSROOMUSEACADEMIC YEAR 196o-61

For the academic year 196o-61 the above mentionedtexts were made available for classroom use. A total ofslightly over roo,000 texts were ordered for this pur-pose. This large total, far exceeding expectations, sotaxed the capacities of the printers producing thesetexts that it proved impossible to do more than supplythese classroom orders. Now, however, we have beenable to reprint these texts in moderate numbers and theyare now available in limited quantities for inspection,and for use in in-service training of teachers, includinguse in summer institutes. An order blank at the end ofthis booklet should be used when ordering any of thesetexts.

ACADEMIC YEAR 1961 -62

Arrangements are now being concluded with theYale University Press for the printing and distributionof SMSG texts. Details of this arrangement togetherwith instructions for ordering these texts for classroomuse for 1961-62 will be contained in a later edition ofthis NEWSLETTER, which will appear in April 1961.

It is not expected that the cost of the SMSG texts forclassroom use in 1961-62 will differ substantially fromthe prices listed at the end of this NEWSLETTER.

NECESSITY OFIN-SERVICE PREPARATION OFMATHEMATICS TEACHERSEXPERIENCE IN SMSGEXPERIMENTAL CENTERS

It should be kept in mind that most secondary schoolteachers, through no faith of their own, were not pro-vided in their pre-service training with the mathematicswhich the use of these texts requires. Consequently;most teachers will need some help, in the form of addi-tional training in mathematics, when teaching these

II

texts for the first time. However, experience in theSMSG Experimental Centers suggests that an in- servicetraining program, taught by a subject matter specialist,either before or during the first year's. use of the texts,will be quite satisfactory in answer to this problem.

Evidence from the experimental centers indicates thatwhen a teacher teaches an SMSG course for the secondtime, the amount of in-service assistance needed and theextra time needed for preparation are both drasticallyreduced, and in many cases disappear entirely. A de-tailed discussion of SMSG's experience with in-serviceassistance to teachers is found in NEWSLETTER No. 5.

An important source of the needed additional mathe-matical training is the program of summer and in-service institutes sponsored by the National ScienceFoundation. A number of the NSF Summer Institutesin 1961 will concentrate heavily on SMSG courses, andmany others will undoubtedly use SMSG materials assupplements to their regular courses.

However, SMSG cannot itself provide any direct as-.sistance to school systems wishing to use SMSG texts.In fact, a basic principle in America is that education islocally controlled and is independent of the FederalGovernment. SMSG, which receives all its financialsupport from the Federal Government through theNational Science Foundation, therefore wishes to donothing which might be interpreted as an attempt toinfluence this local control of education. SMSG willconfine its activities to the preparation and testing ofimproved texts. The decision to adopt these texts, andthe implementation of the decision, is entirely up to thelocal school systems.

A final comment is that there already is considerableevidence that adoption of SMSG texts is an almost ir-reversible process. Most teachers are very reluctant toreturn lo conventional textbooks after once using.SMSG texts.

12

ADVISORY COMMITTEE

A. A. Albert, University of ChicagoF. B. Allen, Lyons Township High School, LaGrange, IllinoisE. G. Begle, School Mathematics Study Group, Yale UniversityLipman Bers, New York UniversityS. S. Cairns, University of IllinoisG. F. Carrier, Harvard UniversityW. L. Duren, Jr., University of VirginiaH. F. Fehr, Columbia UniversityH. V. Funk, Columbia UniversityA. M. Gleason, Harvard UniversityJ. H. Hiavaty, Dewitt Clinton High School, New YorkP. S. Jones, University of MichiganL C. Lay, Orange County State CollegeK. 0. May, Carleton CollegeJ. R. Mayor, American Association for the Advancement of

ScienceA. E. Meder, Jr., Rutgers UniversityE. E. Moise, Harvard UniversityP. M. Naghdi, University of California, BerkeleyR. S. Pieter, Phillips Academy, Andover, MassachusettsG. B. Price, University of KansasR. E. K. Rourke, Kent School, Kent, ConnecticutA. W. Tucker, Princeton UniversityHenry Van Engen, University of WisconsinA. D. Wallace, Tulane UniversityE. L. Walters, William Penn Senior High School, York, Penn-

sylvaniaMarie S. Wilcox, Thomas Can Howe High School, Indian-


New York


J. A. Brown, University of DelawareLenore John, Laboratory School, University of ChicagoB. W. Jones, University of ColoradoP. S. Jones, University of MichiganJ. R. Mayor, American Association for the Advancement of

ScienceP. C. Rosenbloom, University of MinnesotaVeryl Schutt, Supervisor of Mathematics, Washington, D.C.


F. B. Allen, Lyons Township High School, LaGrange, IllinoisEdwin C. Douglas, The Taft School, Watertown, ConnecticutD. E. Richmond, Williams College, Williamstown, Massachu-

settsC. E. 'Bekaa, Yale University -

Henry Swain, New Trier Township High School, Winnetka,Illinois

R. J. Walker, Cornell University

13

PANEL ON MONOGRAPHS

Lipman Ben, New York University, New YorkH. S. M. Coxeter, University of Toronto, Toronto, Ontario,

CanadaP. R. Halmos, University of ChicagoJ. H. Hiavaty, Dewitt Clinton High School, New YorkN. Jacobson, Yale UniversityR. S. Pieter, Phillips Academy, Andover, MassachusettsH. 0. Pollak, Bell Telephone LaboratoriesGeorge Polya, Stanford UniversityH. E. Robbins, Columbia UniversityN. E. Steenrod, Princeton UniversityJ. J. Stoker, New York UniversityLeo Zippin, Queens College, Flushing, New York


J. B. Adkins, Phillips Exeter Academy, Exeter, New HampshireH. F. Fehr, Columbia UniversityJ. L. Kelley, University of CaliforniaL C. Lay, Orange County State CollegeK. O. May, Carleton CollegeB. E. Meserve, Montclair State College, Upper Montclair, New

JerseyG. S. Young, Tulane University

PANEL ON UNDERDEVELOPED MATHEMATICALTALENT

V. H. Haag, Franklin and Marshall CollegeMildred Kehler, Cincinnati Public Schools, Cincinnati, OhioOscar Schaaf, South Eugene High School, Eugene, OregonM. A. Sobel, Montclair State College, Upper Montclair, New

JerseyMarie S. Wilcox, Thomas Carr Howe High School; Indianapolis,

IndianaA. B. Willcox, Amherst College

PANEL ON ELEMENTARY SCHOOL MATHEMATICS

E. Glenadine Gibb, Iowa State Teachers College, Cedar Falls,Iowa

W. T. Guy, University of TexasS. B. Jackson, University of MarylandIrene Sauble, Detroit Public School;M. it Stone, University of ChicagoJ. F. Weaver, Boston University(The Director is ex officio a member of each Panel)

PARTICIPANTS IN THEWRITING SESSIONS

Below are listed the names of all those who partici-pated in one or more uf the SMSG writing sessions. Itwill be observed that the participants included both ex-perienced high school teachers and distinguished re-search mathematicians. Indeed, it is a fundamental beliefof the School Mathematics Study Group that substantial im-provement in the school mathematics curriculum can resultonly from cooperation, on an equal basis, of both these groups.

The mere existence of the sample textbooks for grades7-12 is ample proof that such cooperation is not onlypossible, but also productive.

H. W. Alexander, Earlham CollegeF. B. Allen, Lyons Township High School, LaGrange, IllinoisR. D. Anderson, Louisiana State UniversityB. H. Arnold, Oregon State CollegeAlexander Beck, Olney High School, Philadelphia, Pennsyl-

vaniaE. F. Beckenbach, University of California at Los AngelesE. G. Begle, School Mathematics Study Group, Yale UniversityPaul Berg, Stanford UniversityEmil.Berger, Monroe High School, St. Paul, MinnesotaArthur Bentham University of OklahomaR. H. Bing, University of WisconsinA. A. Blank, New York UniversityShirley Boselly, Franklin High School, Seattle, WashingtonTruman A. Botts, University of VirginiaJ. A. Brown, University of DelawareK. E. Brown, Department of Health, Education and Welfare,

Washington, D.C.J. M. Calloway, Caricton College.Hope Chipman, University High School, Ann Arbor, MichiganR. R. Christian, University of British ColumbiaR. J. Clark, St. Paul's School, Concord, New HampshireMildred Cole, East Aurora Junior High School, Aurora, IllinoisB. H. Colvin, Boeing Scientific Research LaboratoriesJ. A. Cooley, University of TennesseeHelen L. Curran, Glenview School, Oakland, CaliforniaP. H. Daus, University of California at Los AngelesR. B. Davis, Syracuse UniversityR. A. Dean, California Institute of TechnologyCharles DePrima, California Institute of TechnologyMary Dolciani, Hunter CollegeEdwin C. Douglas, The Taft School, Watertown, Connecticut

-~Floyd Downs, East High School, Denver, ColoradoE. A. Dudley, North Haven High School, North Haven, Con-

necticutLincoln Durst, The Rice Institute

S

Florence Elder, West Hempstead High School, West Hemp-stead, New York

Marion Epstein, Educational Testing Service, Princeton, NewJersey

W. E. Ferguson, Newton High School, Newtonville, Massachu-setts

N. J. Fine, University of PennsylvaniaWalter Fleming, Hamline UniversityJoyce D. Fontaine, North Haven High School, North Haven,

ConnecticutF. L. Friedman, Massachusetts Institute of TechnologyHelen Garstens, University of MarylandEsther Gassett, Claremore High School, Claremore, OklahomaH. M. Gehtnan, University of BuffaloRonald Genise, Brentwood Junior High School, Brentwood,

New YorkR. K. Getoor, University of WashingtonE. Glenadine Gibb, Iowa State Teachers CollegeR. A. Good, University of MarylandW. T. Guy, University of TexasV. H. Vaag, Franklin and Marshall CollegeLeon Haand, Kenwood School, Minneapolis, MinnesotaAlice Hach, Racine Public Schools, Racine, WisconsinClarence E. Hardgrove, Northern Illinois UniversityRoyce Hargrove, Corpus Christi Public Schools, Corpus Christi,

TexasR. R. Hartman, Edina-Morningside Senior High School, Edina,

MinnesotaM. H. Heins, University of IllinoisSally Herriot, Cubberly Senior High School, Palo Alto, Cali-

forniaEdwin Hewitt, University of WashingtonArthur A. Hiatt, Santa Clara High School, Santa Clara, Cali-

forniaMartha Hildebrandt, Proviso Township High School, May-

wood, IllinoisThomas J. Hill, Oklahoma City Public School, Oklahoma City,

OklahomaMax Hosier, Iowa State Teachers CollegeLucille Houston, Racine Public Schools, Racine, WisconsinHelen Hughes, Theodore Roosevelt Junior High School, Eugene,

OregonS. B. Jackson, University of MrylandHenry G. Jacob, LotuNiana State UniversityLenore John, Laboratory School, University of ChicagoDonovan Johnson, University of MinnesotaB. W. Jones, University of ColoradoP. S. Jones, University of MichiganPaul S. Jorgenson, Carleton CollegeR. C. Jurgensen, Culver Military Academy, Culver, IndianaH. T. Karnes, Louisiana State UniversityM. L. Keedy, University of MarylandMildred Keiffer, Cincinnati Public Schools, Cincinnati, OhioDavid H. Knowles, Samuel Ayer High School, Milpitas, Cali-

forniaC. J. Koutsopoulos, Educational Testing Service, Princeton, New

JerseyJoseph Lehner, Michigan State UniversityMarguerite Lehr. Bryn Mawr CollegeKenneth Lcisenring, University of Michigan

16

Howard Levi, Columbia UniversityEmma M. Lewis, Washington Public Schools, Washington,

D.C.Eunice Lewis, Laboratory High School, University of OklahomaM. A. Linton, William Penn Charter School, Philadelphia,

PennsylvaniaWilliam G. Lister, State University of New York, Oyster Bay,

New YorkA. E. Livingston, University of WashingtonL H. Loomis, Harvard UniversityNick Lovdjieff, Anthony Junior High School, Minneapolis,

MinnesotaR. V. Lynch, Phillips Exeter Academy, Exeter, New HampshireLeila M. Maneely, Springer School, Los Altos, CaliforniaHelen Marston, Douglass CollegeLysle C. Mason, Phillips University, Enid, OklahomaJ. R. Mayor, American Association for the Advancement of

ScienceMary McDermott, Mt. Diablo Unified School District, Con-

cord, CaliforniaW. K. McNabb, Hockaday School, Dallas, TexasFrances 5. Mettler, Walter Hays Elementary School, Palo Alto,

CaliforniaS. S. Meyers, Educational Testing Service, Princeton, New

JerseyK. G. Michaels, North Haven High School, North Haven,

ConnecticutMuriel Mills, Hill Junior High School, Denver, ColoradoE. E. Moise, University of MichiganE. P. Northrop, University of ChicagoMax Peters, Wingate High School, Brooklyn, New York0. J. Peterson, Kansas State Teachers College, Emporia, KansasB. J. Pettis, University of North CarolinaR. S. Pieters, Phillips Academy, Andover, MassachusettsC. F. Pinzka, University of CincinnatiH. 0. Pollak, Bell Telephone LaboratoriesWalter Prenowitz, Brooklyn CollegeG. B. Price, University of KansasA. L. Putnam, University of ChicagoPenis 0. Redgrave, Norwich Free Academy, Norwich, Con-

necticutMina Rees, Hunter CollegeD. E. Richmond, Williams CollegeC. E. Rickart, Yale UniversityP. C. Rosenbloom, University of MinnesotaElizabeth Roudebush, Seattle Public Schools, Seattle, Wash-

ingtonHarry Ruderman, Hunter College High School, New York CityGeorge Schaefer, Alexis I. Dupont High School, Wilmington,

DelawareHelen A. Schneider, Oak School, LaGrange, IllinoisVeryl Schult, Washington Public Schools, Washington, D.C.J. T. Schwartz, New York UniversityWilla J. Sessions, Hillsborough County Public Schools, Tampa,

FloridaLeila E. Sharp, Alice Robertson Junior High School, Muskogee,

OklahomaRose Mary Shea, Edith C. Baker School, Brookline, Massachu-

settsA. L. Shields, University of Michigan

17

M. A. Sobel, Montclair State College, Montclair, New Jersey0. E. Stanaitis, St. Olaf CollegeRobert Starkey, Cubberlcy High School; Palo Alto, Cali-

forniaWarren Stenberg, University of MinnesotaRothwell Stephens, Knox CollegePhillip Stucky, Roosevelt High School, Seattle, WashingtonHenry Swain, New Trier Township High School, Winnctka,

lllinoisHenry Syer, Kent School, Kent, ConnecticutG. B. Thomas, Massachusetts Institute of TechnologyWesley Thompson, Detroit Public Schools, Detroit, MichiganGeorge M. Truscott, Wilbur Junior High School, Palo Alto,

CaliforniaA. W. Tucker, Princeton UniversityH. E. Vaughan, University of IllinoisJohn Wagner, University of TexasRay Walch, Westport Public Schools, Westport, ConnecticutR. J. Walker, Cornell UniversityA. D. Wallace, Tulare UniversityWilliam C. Walsh, Saddle Brook High School, Saddle Brook,

New JerseyE. L. Walters, William Penn Senior High School, York, Penn-

sylvanizMorgan Ward, California Institute of TechnologyTed Wassam, Ventura School, Palo Alto, CaliforniaF. C. Watson, East High School, Rochester, New YorkJ. Fred Weaver, Boston UniversityG. C. Webber, University of DelawareWarren White, North High School, Sheboygan, WisconsinD. V. Widder, Harvard UniversityMarie S. Wilcox, Thomas Carr Howe High School, Indian-

apolis, IndianaA. B. Willcox, Amherst CollegeWilliam Wooton, Pierce Junior College, Woodland Hills,

CaliforniaJohn E. Yamelle, Hanover College, Hanover, IndianaJ. H. Zant, Oklahoma State University

NOTE: The institutions listed above are those with which the par-ticipants were associated during the writing sessions.

PUBLICATIONS AVAILABLE

TO ORDER, USE THE FORM ON THE INSIDE BACK COVER AND

SEND TO School Mathematics Study GroupBox 2029 Yale StationNew Haven, Connecticut


VOLUME I. SOME BASIC MATHEMATICALCONCEPTS

by R. D. LUC.E

An exposition of elementary set theory, together withillustrations of the use of set concepts in various parts ofmathematics.

Price: Si=

VOLUME 2. EUCLIDEAN GEOMETRY BASED ONRULER AND PROTRACTOR AXIOMS (revisededition)

by C. W. CURTIS, P. H. DAUS, and R J. WAIXER

A study, written for high school teachers, of theapproach to Euclidean Geometry used in the SMSGtenth-grade textbook.

Price: $x.00

VOLUME 3. STRUCTURE OF ELEMENTARYALGEBRA

by VINCENT H. HAAG

An explanation of the approach to algebra foundin the SMSG text "First Course in Algebra" emphasiz-ing the foundations of the subject and the structuralproperties of elementary algebraic systems.

Price: $ Loo

VOLUME 4. GEOMETRY

by B. V. Kuruzov

A translation of a Russian text for teachers. Longerthan other volumes in this series (576 pages), it willprobably be used mostly as a source of supplementarymaterial.

Price: $5.00

19

VOLUME 5. CONCEPTS OF INFORMALGEOMETRY

by RICHARD D. ANDERSON

A study of basic ideas, concepts, and points-.)&viewof geometry, intended primarily for junior high schoolteachers.

Price: $r.00


VOLUME I (revised edition)AND

VOLUME R (preliminary edition)

Traditional mathematics courses for grades 7 and 8indude a review of the operations with whole numbers,fractions, and decimals. Percent is introduced, usuallyin terms of the three cases of percents, each of which istreated separately after various manipulations with per-cents, including fractional and decimal equivalents ofpercents. The traditional courses also have rather ex-tensive treatments of percent applications such as corn -mission, simple interest, discount, and insurance. Astudy of measurement has had an important place, butagain much of this is a review of work done in earliergrades and little or none of it new from a mathematicalpoint of view.

While the new SMSG courses provide for review ofthe fundamentals of arithmetic, this review has beenplaced in a new setting with emphasis on number sys-tems. Number systems are treated from an algebraicviewpoint not only to deepen the student's understand-ing of arithmetic but also to prepare him for the algebrawhich is to come. The work on fractions is introducedby defining a fraction as a numeral for the rationalnumber f, such that b(i) = a, b * o. The grade 8 textstarts with an informal treatment of coordinates andequations, and includes a brief introductory chapter onprobability. Some of the probability problems werewritten by biologists associated with the BiologicalSciences Curriculum Study, and in a chapter on thelever an attempt has been made to use language con-sistent with that of the Physical Science Study Com-mittee physics course. Percent applications have a placein the new courses, as do ether social applications, for

20

example through governmental statistics in the chapteron graphs and in probability.

The new courses give more than one-third of the timeto geometry, which is a very considerable change inemphasis from the traditional. Geometric ideas are in-troduced, first of all, from a non-metric point of viewand then, after a careful treatment of measurement,students are led gradually to a study of properties oftriangles and other geometric figures, plane and solid,through an informal deductive approach. Although

. there is no attempt to give a system of postulates for thegeometry, properties are identified on an intuitive orinductive basis and then these properties are used todraw condusions about, or to prove, other properties.In the chapter on drawings and constructions, instru-ments in addition to the classical ones are introducedand the student is also provided with experience insketching figures, especially three-dimensional figures.A grade 8 chapter on non-metric geometry whichcomes just before the study of t& measurement ofvolumes and surface areas is, in its topological approach,one of the greatest innovations.

VOLUME I: CHAPTER HEADINGS

z. What Is Mathematics?2. Numeration3. Whole Numbers4. Non-Metric Geometry5. Factoring and Primes6. The Rational Number System7. Measurement8. Area, Volume, Weight, and Time9. Ratios, Percents, and Decimals

to. Parallels, Parallelograms, Triangles, and RightPrisms

Circles12. Mathematical Systems13. Statistics and Graphs14. Mathematics at Work in Science

Prices: VOLUME I. Student 82.50VOLUME I. Commentary $2.50

2I

VOLUME II. CHAPTER HEADINGS

T. Rational Numbers and Coordinates2. Equations3. Scientific Notation, Decimals, and the Metric Sys-

tem4. Drawings and ConstructionsS. Symmetry, Congruence and the Pythagorean

Property6. Real Numbers7. Permutations and Selections8. Probability9. Similar Triangles and Variation

1o. Non-Metric Geometry1. Volumes and Surface Areas

12. The Sphere13. Relative Error

Prices: VOLUME II. Student $2.50VOLUME II. Commentary $2.50

MATHEMATICS FOR HIGH SCHOOL

FIRST COURSE IN ALGEBRA (revised edition)

The SMSG ninth grade course is based upon structureproperties of the real number system. This developmentof algebra is interesting, meaningful, and mathematicallysound. It helps bring out the nature of mathematics andstrengthens the student's algebraic techniques by re-lating them to basic ideas. Definitions and propertiesare carefully formulated and there is some work withsimple proofs. The reading material, which is an im-portant part of the course, is designed to help the studentdiscover ideas. The number line and the simpler lan-guage of sets are used to help express the ideas. In-equalities are treated along with equations.

As its title suggests, the SMSG ninth grade coursecovers essentially the same material as does a con-ventional first year algebra text. It teaches the studenthow to perform the fundamental operations with real'numbers and with variables and how to do the usualalgebraic manipulations, including factoring, powersand roots, and work with polynomials and fractional ex-pressions. It shows how to solve equations up throughquadratic equations in one variable and linear equationsin two variables..Graphs oflinear and quadratic functionsare treated. There is much experience in solving wordproblems.

22

CHAPTER HEADINGS

x. Sets and the Number Line2. Numerals and Variables3. Sentences and Properties of Operations4. Open Sentences and English Phrases5. The Real Numbers6. Properties of Addition7. Properties of Multiplication8. Properties of Order9. Subtraction and Division for Real Numbers

to. Factors and Exponentsit. Radicals12. Polynomial and Rational Expressions13. Truth Sets of Open Sentences14. Graphs of Open Sentences in Two Variables15. Systems of Equations and Inequalities16. Quadratic Polynomials17. Functions

Prices: First Course in Algebra. Student $2.50Firs: Course in Algebra. Commentary $2.50

GEOMETRY (revised edition)

The SMSG geometry text differs from conventionalones in content, postulational scheme, and manner oftreatment.

1. No artificial distinction is made between plane andsolid geometry, and a considerable amount of the latteris included. Also, an introduction to analytic planegeometry is provided.

2. The postulate system is a modification of Birk-hofrs, and is complete. Real numbers are used freelythroughout the text, both in the theory and in problems.

3. Accuracy in the statement and use of postulates,definitions, and theorems is emphasized.

On the other hand, the text is still basically a treat-ment of synthetic Euclidean geometry, covering theusual topics: congruence, similarity, parallelism andperpendicularity, area, circles, and construction withstraight-edge and compass. There is a main sequence ofproved theorems, some minor stated theorems withproofs left as exercises for the students, and a long listof "originals." The basic postulates, definitions, andtheorems are motivated by appeal to intuition, and manypractical and computational problems are given.

23

CHAPTER HEADINGS

I. Common Sense and Organized Knowledge2. Sets, Real Numbers and Lines.3. Lines, Planes and Separation4. Angles and Triangless. Congruences6. A Closer Look at Proof7. Geometric Inequalities8. Perpendicular Lines and Planes in Space9. Parallel Lines in a Plane

io. Parallels in Spaceii. Areas of Polygonal Regions12. Similarity13. Circles and spheres14. Characterization of Sets. Constructionsis. Areas of Circles and Sectors16. Volumes of Solids17. Plane Coordinate Geometry

Prices: Geometry. Student $2.soGeometry. Commentary $2.so

INTERMEDIATE MATHEMATICS (revised edition)

The SMSG eleventh grade text differs from traditionaltexts in the following important ways:1. The SMSG text makes much greater demands on

the student's ability to learn by reading carefully wordedexpositions. The writers believe that the develop-ment of this ability is essential for success in collegemathematics.

2. The study of number systems is stressed as the basisfor all understanding of both elementary and ad-vanced mathematics.

3. The idea that algebra is a logical structure built on arelatively small number of fundamental principles isemphasized throughout the text.

4. Presentations which lead the student to make cer-tain pre-determined "discoveries" are used where ap-propriate.

5. Proof is emphasized throughout in order that thestudent may gain some idea of the nature of a validmathematical argument.

6. The function concept is developed spirally throughoutthe text.

7. Coordinate geometry is introduced earlier than usualand is used as a tool in the development of subse-quent sections, notably those on trigonometry.

8. The presentation of logarithms reflects contempo-rary usage which requires more understanding oflogarithms and exponential functions and relativelyless emphasis on logarithmic computations.

9. The treatment of trigonometry emphasizes identi-ties, equations, and graphs of the trigonometric func-tions more than the computations required in thesolution of triangles.

io. Vectors are developed as a mathematical system andare applied to the solution of a wide variety ofproblems.

The writers hope that through the studying of thistext the student will acquire some ability to handle un-foreseen and unforeseeable problems.

The SMSG eleventh grade text is similar to con-ventional texts in these respects.

z. The text begins with a review and extension of thebasic skills of first year algebra. This review is in-cluded in the initial study of number systems.

2. The content is essentially the same as that found inconventional courses in trigonometry and collegealgebra.

3. Practical applications are given about the sameamount of attention as in conventional texts. It wasnot possible to increase the number of applicationsappreciably without making unwarranted assump-tions about the student's understanding of relatedfields.

4. The exposition makes use of many illustrative ex-amples and drawings.

s. There is an abundant supply of exercises which havebeen carefully graded as to difficulty.

CHAPTER HEADINGS

t. Number Systems2. An Introduction to Coordinate Geometry in the

Plane3. The Function Concept and the Linear Function4. Quadratic Functions and Quadratic Equations5. Complex Number System6. Equations of the First and Second Degree in Two

Variables7. Systems of Equations in Two Variables8. Systems of First Degree Equations in Three Vari-

ables9. Logarithms and Exponents

to. Introduction to TrigonometryI1. The System of Vectors12. Polar Form of Complex Numbers13. Sequences and Series14. Permutations, Combinations, and the Binomial

Theorem15. Algebraic StructuresPrices: Intermediate Mathematics. Student $2.56

Intermediate Mathematics. Commentary $2.50

ELEMENTARY FUNCTIONS (revised edition)

The subject matter of Elementary Functions is basicallyconventional. It includes such topics from the theory ofequations as the remainder and factor theorems and theusual methods for finding rational roots. The studentwill find the laws of exponents and logarithms and therules for changing the base. The chapter on circularfunctions contains the familiar addition and subtractionformulas and their consequences, identities and equa-tions, and inverse trigonometric functions. An appendixtreats the solution of triangles. Emphasis is laid ongraphs. However, each of these topics is treated withsome novelty and in a new spirit.

Elementary Functions applies the concept of mappingto polynomial, exponential, logarithmic, and circularfunctions. Effective use is made of the ideas of composi-tion and inversion. The treatment of tangents is intuitive,elementary, and rigorous. It permits the introduction of.Newton's method, and applications to maximum-mini-mum problems and to graphing. The treatment preparesfor calculus without trespassing urn it. The explanation

26

of exponentials and logarithms is novel and urusuallyclear and thorough. Trigonometry is freshly developedin line with the mapping idea. The style is informal.Explanations are full and concrete, and they convey thespirit of mathematical thinking.

CHAPTER HEADINGS

I. Functions2. Polynomial Functions3. Tangents to Graphs of Polynomial Functions4. Exponential and Logarithmic Functions5. Circular Functions

Prices: Elementary Functions. Student $2.00Elementary Functions. Commentary $2.00

INTRODUCTION TO MATRIX ALGEBRA (re-vised edition)

This text is designed for the last half of the lathgrade. It is devoted to a study of matrices, including ap-plications to solutions of systems of linear equationsand to geometry. At the same time careful attention isdevoted to algebraic structure, but not to the pointwhere a barren presentation results. Mathematics is in-troduced which is new to the student and the structureis developed as the text proceeds. It is the intent of thetext to put the student close to the frontiers of mathe-matics and to provide striking examples of patterns thatarise in the most varied circumstances. It provides aneffective language and some dynamic concepts that willprove useful in many college courses. A special set of"Research Exercises" is appended in the hope that somestudents may be introduced to real mathematical re-search.

CHAPTER HEADINGS

T. Matrix Operations2. The Algebra of a X 2 Matrices3. Matrices and Linear Systems4. Representation of Column Matrices as Geometric

Vectors5. Transformations of the PlanePrices:

Introduction to Matrix Algebra. Student $1.50Introduction to Matrix Algebra. Commentary 11 1.5o

27

SUPPLEMENTARY MATERIALS

MATHEMATICS FOR THE JUNIOR HIGHSCHOOL SUPPLEMENTARY UNITS (revised edi-tion)

CHAPTER HEADINGS

r. Sets2. Special Figures in Projective Geometry3. Repeating Decimals and Tests for Divisibility4. OPen and Closed Paths .

S. Finite Differences6. Recent Information on Primes7. Games

Prices: Mathematics for Junior High SchoolSupplementary Units. Student $ .75Mathematics for Junior High SchoolSupplementary Units. Commentary $ .75

ESSAYS IN NUMBER THEORYThese supplements were written for students who are

especially good in mathematics and who have a livelyinterest in the sublet/. The author's aim in (I) and (a)is to lead the reader to discover for himself some in-teresting results and to experience the thrill of mathe-matical discovery. The others are more expository innature, but they contain exercises to clarify the materialand to give the reader a chance to work with the con-cepts which are introduced.

VOLUME I: CHAPTER HEADINGSt. Prime Numbers2.. Congruence3. The Fundamental Theorem of Arithmetic

Price: $ .25

VOLUME II: CHAPTER HEADINGST. Arithmetic Functions I2. Arithmetic Functions II3. Arithmetic Functions III4. The Euclidean Algorithm and Linear Diophantine

Equationss. The Gaussian Integers6. Fermat's Method of Infinite DescentT. Approximation of Irrationals by Rationale8. A New Field

Price: $ .50

MATHEMATICS FOR HIGH SCHOOLINTERMEDIATE MATHEMATICS (PART I)SUPPLEMENTARY UNIT I

(THE DEVELOPMENT OF THE REAL NUMBERSYSTEM) (revised edition)

This is a revision of Chapter I "The Real NumberSystem," INTERMEDIATE MATHEMATICS (pre-liminary edition), Part I.

Price: $ .75

CONFERENCE REPORTS

REPORT OF A CONFERENCE ONELEMENTARY SCHOOL MATHEMATICS

This publication includes abstracts and resumes of thepresentation, discussions, and recommendations of aconference on elementary school mathematics held onFebruary 13 and 14, 1959. The participants includedcollege and university mathematicians, high schoolteachers, educational experts with special interest inarithmetic, supervisors, elementary school teachers, psy-chologists, and representatives of scientific and govern-ment organization having an interest in mathematics.

Price: $ .p

REPORT OF AN ORIENTATION CONFERENCEFOR SMSG EXPERIMENTAL CENTERS

This publication includes abstracts of the presentationsand discussions of a conference on mathematics forgrades 7-12 held on September 19, 1959. The con-ference was called so that the teachers and consultantsin the experimental centers for 1959-60 could discussthe aims, objectives, and content of the SMSG booksfor grades 7-12. The participants included a representa-tive sampling of the text authors.

Price: $

29

REPORT OF AN ORIENTATION CONFERENCEFOR SMSG ELEMENTARY SCHOOLEXPERIMENTAL CENTERS

This publication includes abstracts of the presentationsand discussions of a conference on elementary schoolmathematics held on September 23-24, 1960. The con-ference was called so that the teachers and consultantsin the elementary school experimental centers for 1960-61 could discuss the aims, objectives, and content of theSMSG materials for grades 4, 5, and 6. These partici-pants included a representative sampling of the textauthors.

Price: $i.00

MATHEMATICS FOR THE ELEMENTARYSCHOOL

. The nature of these materials was described on page7. Grade placement for the units listed is described ina preface to the volume in which all of these units havebeen collected. These materials are not yet ready forgeneral classroom use and are being made available atthis time to inform the mathematical community ofthe present state of the SMSG project on elementarymathematics. The two volumes, pupil's text and teacher'scommentary, are available ONLY as a single unit. Theycannot be purchased separately.

UNIT HEADINGS

i. Concepts of Sets2. Numeration3. Nature and Properties of Addition and

Subtraction4. Techniques of Addition and Subtractions. Sets of Points5M. Sets of Points6. Recognition of Common Figures7. Nature and Properties of Multiplication and

Division8. Techniques of Multiplication and Division9. Developing the Concept of Fractional Numbers

to. Linear MeasurementIt. Factors, Primes, and Common Denominators12. Properties and Techniques of Addition and. Sub-

traction of Fractional Numbers

30

13. Side and Angle Relationships of Triangle14. Measurement of Angles15. Extending Systems of Numeration17. Area19. Multiplication and Division of Fractional Num-

bers21. Area of Rectangu ar Regions24. Introducing Exponents

Price: Sio.00

Books listed on the order blank and described inpages 19 through 3o are the only books available at thistime. Supplies of earlier versions of these materials havebeen exhausted.

In ordering books please use the order blank enclosedin this NEWSLETTER or specify the exact tide in full whenordering.

As indicated on page t t the current supply of ma-terials is limited. Please do not submit orders for CLASS-ROOM quantities for the academic year 1961-62 atthis time. Texts for grades 7 through 12 will be availableat approximately the same price for classroom usethrough the Yale University Press. The April issue ofthe SMSG NEwsLETTER (No. 7) will contain full par-ticulars.

If you are not now on our mailing list but wish toreceive further issues of this NEwscErrEa, please request,by means of a post card, that your name be added tothe mailing list.

31

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41

SCHOOLMATHEMATICSSTUDY GROUP

Newsletter No. 11March 1962

tm

CONTENTS

THE FUTURE or SMSG, 3

BYLAWS, 4

MEMBERSHIP: ADVISORY BOARD AND PANELS, 7

NEW PROJECTS, 10

NEW PUBLICATIONS, 14

DISTRIBUTION OF SMSG PUBLICATIONS, 21

ORDER FORM, center

"PERMISSION TO REPRODUCE THIS COPY-RIGHTED MATERIAL HAS SEEN GRANTED BY

E. G. BegleTO ERIC AND ORGANIZATIONS OPERATINGUNDER AGREEMENTS WITH THE NATIONAL IN-

snruTE OF EDUCATION. FURTHER REPRO-DUCTION OUTSIDE THE ERIC SYSTEM RE"QUIRES PERMISSION OF THE COPYRIGHTOWNER."

1962 by The Board of Trustees of theLeland Stanford junior UniversityAU rights reserved

Printed in the United States of America

THE FUTURE OF SMSG

When the School Mathematics Study Groupcame into being on March 1, 1958 it was notclear how long it would take to accomplish itsobjectives. Undoubtedly many of those who par-ticipated in the creation of SMSG felt that abrief period of intensive effort would be sufficient.-

A few years of experience, however, were. enough to make it clear that the general kindsof activities undertaken by SMSG, in particularthe close collaboration between classroom teach-ers and research mathematicians, ought to becontinued indefinitely. The completion of severalmajor projects gives evidence of the value ofsuch collaboration; in the course of completingthem the need for several new projects developed.

For this reason a conference was held in Feb-ruary 1961 to consider ways of continuing thework which SMSG had begun. The sixty partici-pants represented all parts of the country andall parts of the mathematical profession, as well asvarious degrees of interest in mathematics educa-tion, ranging from full-time participation in cur-riculum projects to passive but interested obsei-vation from the sidelines. The main recommen-dation resulting from this conference was thatSMSG should continue in existence, carrying onwith its present programs and, in addition, broad-ening its scope to provide leadership in all areasof school mathematics education.

In addition, it was recommended that the Ad-visory Committee, which sets the basic policyfor the School Mathematics Study Group, begiven a more formal structure and that, in par-ticular, some members of the Advisory Commit-tee be named by the Conference Board of the'Mathematical Sciences, which represents all the

major mathematical organizations in the UnitedStates.

These recommendations were accepted and inOctober, 1961, the Advisory Committee (nowrenamed the Advisory Board) adopted a set ofbylaws embodying these recommendations. Thesebylaws follow.

BYLAWS

I. PURPOSES

The primary purpose of the SMSG is to fosterresearch and development in the teaching ofschool mathematics. This research will consist, inpart, of a continual review of the mathematicscurricula in the schools, as an aid in the selectionand design of promising experiments. It will alsoconsist in part of an analysis of the results ofexperimental teaching, as an aid in judgingwhether the objectives of various programs arebeing achieved. But the work of the SMSG shouldconsist primarily in the development of courses,teaching materials and teaching methods.

A great variety of these are needed. In thefirst place, the range of ability among the stu-dents who ought to be learning mathematics isso wide that special provisions to be made,neefAlyso that the students at various abili levek canbe taught in appropriate styles and it appropri-ate paces. Moreover, there should be bold experi-ments, with courses differing sharply from pres-ent practice in their style, or their content, orboth; if experimentation is limited to programswhose desirability is obvious and whose feasi-bility is predictable, then some of the best oppor-tunities are likely to be missed. Sources of othersuggestions for needed activities and projectsare: (a) the declaration of aims and purposesof the SMSG, July, 1958; (b) the actual activitiesand productions of the SMSG, 1958-1961; ( c) theRecommendations for the Conference on FutureResponsibilities for School Mathematics, Febru-ary 1961.

It is a part of SMSG's task, in cooperationwith the several mathematics organizations, toencourage exploration of the hypotheses under-lying mathematics education. Care should betaken to give attention to the needs of otherdisciplines interested in the mathematical prepa-ration of students. It is also part of the SMSG'stask to publicize its own work and make thiswork available to people who might use it. Itshould be understood, however, that the basicjob of the SMSG is not to defend any orthodoxy,old or new, by passing judgment on questions ofeducational policy, but rather to make contribu-tions to the data on which such judgments mustbe based. In this spirit, efforts should be madeto help teacher-education institutions to offerprograms which will enable their graduates toteach the new courses in the schools. The SMSGshould also explore the possibility of promotingthe developments of in-service education ofteachers and supervisors as a continuing andnecessary part of their professional lives.

The success of the SMSG's enterprise dependson the full participation of mathematicians fromthe colleges and universities and classroom teach-ers from the schools. It is a fundamental policythat the work of the SMSG be collaborative inthis sense.

II. ORGANIZATION

The executive officer of the SMSG shall be theDirector. There shall also be an Advisory Boardand an Executive Committee.

The first Advisory Board of the SMSG shallconsist of

(1) The. Director, ex officio,

(2) Eight members of the SMSG AdvisoryCommittee ( as constituted in the Springof 1961), to serve for two years,

(3) Eight members of the same body,serve for one year,

(4) Four members to be elected by the samebody in the Fall of 1961, to serve forthree years, and

( 5) Four members to be elected by theConference Board of. the MathematicalSciences, to serve for three years

Thereafter, the term of each member shall bethree years, beginning on September 1 and end-ing on August 31; and no member shall servetwo consecutive terms.

Each year, four new members shall be ap-pointed by the Conference Board of the Mathe-matical Sciences and four shall be elected by theAdvisory Board. Vacancies due to uncompletedterms shall be filled by the Executive Committee.

The Advisory Board shall elect ( 1 ) its ownchairman, who shall also be the chairman of theExecutive-Committee, and-(2)-three-other-mem--hers of the Executive Committee. All of theseofficers shall serve for one-year terms, beginningon September 1 and ending on August 31.

The Executive Committee shall consist of itsfour elected members, together with the` Director,ex officio. If the directorship becomes vacant, anew Director shall be appointed by the Execu-tive Committee.

The chairman of the Advisory Board shall pre-side at all meetings of the Advisory Board andthe Executive Committee. Each year he shallappoint a nominating committee, which shallmake nominations for all of the annual electionsby the Adirisory Board. At the meeting at whichthese elections are held, nominations from thefloor shall- he permitted.

Meetings of the. Advisory Board and the Exec-utive Committee shall be called by the. Director:The Director shall also appoint such ad hoc com-

..rnittees, panels and writing teams as may be.needed, and-prepare-reportg- on SMSG activitieson request- of the Advisory Board.

Amendments to the Bylaws maybe be made bthe Advisory Board. .

MEMBERSHIPADVISORY BOARDPANELS

ADVISORY BOARD 1961-62

Terms Expire September 1,1962

Lipman Bers, New York UniversityS. S. Cairns, University of Illinois

W. L. Duren, University of VirginiaL. C. Lay, Orange County State College, Fullerton,

CaliforniaA. E. Meder, Jr., Rutgers UniversityG. B. Prim, Conference Board of the Mathematical

ScicncesH. Van Engen, University of WisconsinE. L. Walters, William Penn Senior High School,

York, Pennsylvania

Terms Expire SeptembefIT1963

F. B. Allen, Lyons Township School, LaGrange, IllinoisA. M. Gleason, Harvard University

Chairman of the Advisory BoardSeptember 1, 1961 - August 31, 1962

P. S. Jones, University of MichiganJ. R. Mayor, American Association for the

Advancement of ScienceR. S. Pieters, Phillips.Academy, Andover, Massachusetts

R. E. K. Rourke, Kent School, Kent, ConnecticutA. W. Tucker, Princeton UniversityMarie S. Wilcox, Thomas Carr. Howe High Schooi,

Indianapolis, Indiana

terms Expire September 1, 1964

Leslie Beatty, Chula Vista City School District,Chula Vista, California

Roy Dubisch, University of WashingtonW. E Ferguson, Newton High School, Newtonville,

MassachusettsG. C. Pimentel, University of California

H. O. Pollak, ilell Telephone Laboratories,Irene Sauble, Detroit Public Schools

H. Stone, University of ChicagoJ. F. Weaver, Boston University

Members of the Executive Committee'Appointed by the Conference Board of theMathema Neal Sciences

PANEL ON 7th AND SA GRADES

J. A. Brown, University of DelawareLenore John, Laboratory School, University of ChicagoB. W. Jones, University of ColoradoP. S. Jones, University of MichiganJ. R. Mayor, American Association for the

Advancement of ScienceP. C. Rosenbloom, University of MinnesotaVeryl Schu lt, Supervisor of Mathematics, Washington, D.C.


F. B. Allen, Lyons Township High School, LaGrange,Illinois

Edwin. C. Douglas, The Taft School, Watertown,C4Necticut

D. E. Richmond, Williams College, Williamstown,Massachusetts

C. E. Rickart, Yale UniversityR. A. Rosenbaum, Wesleyan UniversityHenry Swain, New Trier Township High School,

Wirmetka, IllinoisR. J. Walker, Cornell University

PANEL-ON-MONOGRAPHSLipman Bers, New York UniversityW. G. Chinn, Portal Junior High School, San FranciscoP. J. Davis, National Bureau of StandardsMark K.sc, The Rockefeller InstituteP. R. Halmos, University of MichiganJ. H. Hlavaty, Dewitt Clinton High School, New YorkR. S. Pieters, Phillips Academy, Andover, MassachusettsN. E. Steenrod, Princeton University


J. B. Adkins, Phillips Exeter Academy, Exeter,New Hampshire

H. F. Fehr, Columbia UniVersityJ. L. Kelley, University of CaliforniaL. C. Lay, Orange County State CollegeK. 0. May; Carleton CollegeB. E: Meserve, Montclair State College, Upper Montclair,

New JerseyG. S. Young, Thiene University

PANEL' ON UNDERDEVELOPED MATHEMATICAL-.TALENT

V. H. Haig, Franklin and Maishall CollegeMildred Keifer, Cincinnati Public Schools, Cincinnati,

' OhioOscar Schaaf; .Sonth Eugene Higgh School, Eugene;Oreien

UppeiMontcliii,New Jersey

.

Marie S. Wilcox,- Themes Carr Howe High S'Indianapolis, Indiana

Amherst College, Amherst, Massachusetts

PANEL ON ELEMENTARY SCHOOLMATHEMATICS

Leslie Beatty, Chula Vista City School District,Chula Vista, California

E. Clenadine Cibb, Iowa State Teachers College,Cedar Falls, Iowa

W. T. Guy, University of TexasS. B. Jackson, University of MarylandIrene Sauble, Detroit Public SchoolsM. H. Stone, University of ChicagoJ. F. Weaver, Boston UniversityR. L. Wilder, University of Michigan

PANEL ON TESTS

R. Alpert, Harvard UniversityMax Beberman, University of IllinoisR. P. Dilworth, California Institute of TechnologyJ. Kagan, The Fels Research Institute, Yellow Springs,

OhioM. A. Linton, Jr., William Penn Charter School,

Philadelphia, PennsylvaniaW. C. Lister. State University of New York

--PANELONPROGRAME01.1A1H4INGR. C. Buck, University of WisconsinE. L. Hammond, Phillips Academy, Andover,

MassachusettsL. D. Hawldnson, San Franc sco Public SchoolsJ. G. Holland, Harvard link fersityW. J. McKeachie, University of MichiganE. E. Moise, Harvard UniversityH. O.-Pollak, Bell Telephone LaboratoriesD. W. Taylor, Yale University

PANEL ON SMALL PUBLICATIONS

B. D. Anderson, Lomisiana State UniversityM. P. Bridgess, Roxbury Latin School, West Roxbury, --

Massachusettsj. M. Calloway, Kalamazoo College, Kalamazoo, MialginR.. J. Clark, St. Paul's School, Concord, New HampshireRoy thibisck University of WashingtonT. H. Hill, Oklahoma City SchoolsK. S. Kalman, Lincoln High School, Philadelphia,

PennsylvaniaAugusta L. Schuster, Iowa State Teachers College,

Ceder Falls, Iowa1 Henry W. Syer, Kent School, Kent, Connecticut

(The Director is ex officio a member of the. Advisory. Board of each Panel). -

NEW PROJECTS

Newsletter No. 6 ( March 1961) listed the variousprojects being carried out by SMSG at that time.Since then, four new projects have been started.

SMALL PUILICATIONS

In this project _two kinds of publications will beprepared. The first- will consist of supplemen-tary material, especially for the more able stu-dents. The two small volumes "Essays in NumberTheory" are typical examples.

The-second-kind-of-publication-will-consist4units, designed for classroom use, which can beused by teachers who would like to introducesome new ideas in their classes but are unable,for one reason or another, to undertake a newfull year course. The SMSG "Junior High SchoolMathematics Units" are a typical example.Presumably some material of this kind can beextracted from present SMSG publications, butthe Panel ( see p. 9) in charge of this project isfree to search elsewhere for suitable materialand also to arrange for the preparation cf newmaterial.

PROGRAMED LEARNING PROJECT

The School Mathematics Study Group hasundertaken a project involving programed learn-ing in mathematics, with special reference, to thekind of mathematics illustrated in the texts al--ready prepared by the SMSG. This project isbeing supervised by a Panel consisting of eightuniversity and secondary school personnel in thefields of mathematics and psychology. ( See listof Panel on Page 9.)

The Panel decided as a first step to have,the,

mathematics in the SMSG First Course inAlgebra

programed. Two program forms were chosen:one consists of a series of constructed responsequestions with the answers provided; the otherconsists of passages followed by multiple choicequestions with corrective material for incorrectalternatives selected.

In preparing these programs, the usual SMSGprocedure will be followed; namely, writing byteams, reviewing, test-teaching, and revising. Theprogram writing and reviewing is being done bya joint effort of mathematicians, teachers ofmathematics, psychologists. These materialswill be tested in a variety of classrooms andinstructional sitialions and revised on the basisof the experience gained. Such materials asprove effective will be made generally available.

TRANSLATION PROJECT

.--This-project--is-devoted-to-translating-into-theSpanish language the sample texts prepared bySMSG for grades seven and nine. It is antici-pated that there will also be a demand forgrades eight and ten.

The primary purpose is to facilitate the mathe-Matics education of the Spanish speaking citizensof the United States, principally in Puerto Rico.The translation into Spanish is essential for theremoval of the language barrier to clarity ofunderstanding.

Two centers are involved in the project. Oneis located at the University of Puerto Rico, theother at Stanford University. The texts will beavailable for use in September, 1982.

Preliminary chapters of the above materialwere presented at the Inter-American Conferenceon Mathematics Education held at Bogota, Co-lombia from December 4-9, 1961 where theyaroused great interest.

The Advisory Panel for this project consistsof Professors E. G. Beg le, Stanford University,H. F. Fehr, Columbia University, Mariano Gar-cis, University of Puerto Rico, and Max Kramer,San Jose State, College.

LONG TERM STUDY

The purpose k.-14. this study is to obtain informa-tion which can be used, on the one hand, byteachers, school administrators, parents, and otherinterested persons in evaluating the new mathe-matics curricula, and, on the other hand, by thosewho may wish to undertake further curriculumimprovements in the future.

In this study students starting at three differentgrade levels will be followed for a period of fiveyears. A large quantity of information on the stu-dents will be collected at the beginning, and theperformance of the students will be measuredfrom time to time, both in mathematics and alsoiri science Courses. More Acally, a group ofstudents will be chosen who will enter fourthgrade in September 1962 in schools where it ishighly likely that their mathematics courles_for__the next five years will be of the kind prepared bySMSC. For reference purposes a similar groupwill be chosen in schools where it is highly likelythat any of the mathematics courses for the nextfive years will be conventional. Similar pairs ofgroups of students will also be chosen at the 7thand 10th grade levels.

In order to factor out individual teacher differ-ences, local school situations, etc., and also to takeinto account the expected drop-out due to movesfrom one school district to another, all of thesegroups must be large and will involve a substan-tial number of schools in various parts of thecountry.

A battery of tests will be administered at thebeginning of the study which will measure apti-tude and achievement in mathematics, attitudes,and primary mental abilities, and such other items

,. as the Panel may decide on.

It is planned to measure achievement and atti-tudes of all students included in the study at theend of the third and fifth years. Selected sampleswill also be measured at the end of the first, sec-ond and fourth years. In addition, selected sameples will be tested for performance in Eience

ORDER FORM

Quantity Title and Description (With Code Numbers)

MATHEMATICS FOR THE ELEMENTARY SCHOOLGRADE 4 Student text (E4-P) and Teacher

Commentaries (CF4 -1P) and CE4-2P) 3 vol. set $500GRADE 5 Student text (E5-P) and Teacher

Commentaries (CE5-1P and CE5-2P) 3 vol. set $5.00GRADE 8 Student text (03-P) and Teacher

Commentaries (CE8-1P and CE8-2P) 3 vol. set $5.00

INTRODUCTION TO SECONDARY SCHOOL MATHEMATICSStudent teats (IS 1-4) and Teac.her Commentaries (CIS 1-4) 8 vol. set $8.00

INTRODUCTION TO ALGEBRAStudent texts (IA 1-4) and Teacher Commentaries (CIA 1-4) .... 8 vol. set $8.00

GEOMEMY--WITH-GOORDINMTStudent texts (GW 1-3) and Teachers Commentaries (CCW 1-3) . 13 vol. set $13.00

Unit/Cost Total

STUDIES IN MATHEMATICSSome Basic MathemaScal Concepts (SM-1) $ 80Euclidean Geometry Based a.! Rulir and Protractor Axioms.... . (SM-2) $ .90Structure of Elementary Algel.:- (SM-3) $1.40Geometry (SM-4) $2.75Concepts of Informal C..eometry

.,;

4`;4

(SM-5) $1 45. : ''''''.1' ,P24

Number Systems (shi:e), .ss46 . '...,-,,:.,2-AIntuitive Geometry' (SM77); $115.. -,1;--i-,=.0*

Concepts of Algebra (SM43) stem -,-.:

JUNIOR HIGH SCHOOL' MATHEMATICS UNITS'Volume 1. NuMber,Syshoni, Studeni. . : . .. ... . . . . .... . (y-Iy.Volnise I: Number System's,- Commeritiny-: . .......".-. . . . . . . .... (OUT].

Velume 2 Geometry, Student , .. (U -2)'Volume. 2. Geometry, bommintary ' (CU -2)Volume 3. A p p l i c a t i o n s , Student :. . . ..- .. %.., (U4)Volume 3. ApplicatiOns, Commentary : : :. .. . . . . . .''. '. '.:: . (CU-3)"PKG. Complete Seta (one each of the above 8 books) -:. ., : . . . -. .1'1

SUPPLEMENTARY UNITSMathematics for Junior `High School Supplcimentary' Unita, Student.. (jsy?-..Mathematics for Junior High SehOol Supplementary Units,

Commentary . . ........... .... . .... : . 1CJSU

Quantity Title and Description (With Code Numbers) Unit /Coat Total

Eisays on Number Theory I (HSU-1) $ .30Essays on Number Theory II (HSU-2) $ 50Development of the Real Number System (HSU-3) $ .90

CONFERENCE REPORTSReport of a Conference on Elementary School Mathematics (CR-1) $ .40Report of an Orientation Conference for SMSG Experimental Centers (CR-2) $1.00Report of an Orientation Conference for SMSG Elementary

School Experimental Centers (CR-3) $1.00

1. All SMSC publications are priced at cost.

2.. Orders for less than $10.00 value accompanied by remittance, will be shipped at list prices postpaid.

3. Orders for $10.00 or more value from accredited schools will be allowed 10% discount but transportation willbe added.

ia-schoobrimd-residentsr please-ailow-for4%-SalerTax7------

INVOICE ORDERADDRESS NOIf this order is to be billed to a school or school dis- SHIPPING ADDRESStrict, please indicate full name and address. Use only if different than invoice address.

ORDERS BY INDIVIDUALS should be by full remittance. Please, give frkil mane and

END .ORDERS AND MAKE CHECKS PAYABLE TO: A. C. 'VROMAN, INC.367 SOUTH PASADENA AVENUEPASADENA, CALIFORNIA

are..

courses each year. Students entering the lathgrade at the beginning of this study will be fol-lowed when they reach college as far as possible.Information on both achievement and career

. choices will be sought.

Both the initial battery of tests and the achieve-ment tests to be used in this study will be plannedand prepared under the supervision of the Panelon Tests, listed on page 9.

NEW PUBLICATIONS

TEXTBOOKS WITH TEACHER COMMENTARIES

Mathematics for the Elementary School[Preliminary Edition]

Grade 4 3 vol. $5.00



(For Grade 4 the pupil text is in one volumeand the teacher commentary in two volumes. Thiss also true for Grade 5 and Grade 6.)

The SMSG materials for grades 4, 5, and 6 pro-vide for review and extentliole operationsof arithmetic which the pupils have had in eachpreceding grade. The study of the fundamentaloperations is extended in the familiar pattern tolarger numbers, fractional numbers and decimals.The mathematics traditionally presented in thesegrades is included in Mathematics for the El&=wary School.

In addition to this the commutative, associa-tive, and distributive properties are introducedearly and used whenever appropriate. Numera-tion systems with different bases are used to clarifythe meaning of place value, to deepen understand-ing of the operations, and to show common pat-terns in the different systems.

Addition and subtraction, as well as multiplica-tion and division, are presented as inverse opera-tions. The set _concept is introduced early withconsiderable practice in the use of the languageof sets. Facility in the use of both concept andlanguage is strengthened in subsequent chapteri','which employ their use

Careful attention is paid to use of language andsymbols throughout the materials. When symbolsfacilitate the presentation they are introdueed.

Adequate experience in problem solving is pro-vided throughout the three grades to emphasizetranslation of relationships in problems into sim-ple mathematical sentences and to establish firmlythe skills required in the operations.

Approximately one-third of the materials in thethree grades is geometry. Concepts from planeAnd solid geometry are presented together. Theapproach is intuitive and informal. Both materialfor the pupil and the suggestions for the teacherencourage the exploratory method of learning; thepupil is encouraged to draw conclusions from hisstudy of many examples. No proofs are given orexpected. Inferences expected of the pupil areleft to be proved in the later work in formal geom-etry. From the use of a segment as a standard unitof length for measuring lengths, the transition toa standard snit of area to obtaining the measure

-plane-regionTand-the-use-of-a-standard-unit---of volume to obtain the measure of the volume ofa three-dimensional figure, proceeds naturally.

Arrangement of topics is influenced both bynecessary sequence and attempt at variety. Grade4 is the beginning point and ideas introducedthere are expanded in subsequent grades. If pupilsare to begin with Grade 5 or Grade 6 they willneed background in special topics provided inthe preceding textbooks.

Introduction to Secondary School Mathematics[Preliminary Edition]Student texts (IS 1-4) and 8 volume setTeacher Commentaries (CIS 1-4) Price $8.00

The Introduction to Secondary School Mathe-matics is an adjusted revision of Mathematics forJunior High School Volumes I and II. Thechanges and adjustments whith were made wereprompted by a desire to simplify the presentationand reduce the reading difficulty. Explanatorysections were shortened and exercises added tolead the pupils through simple steps to appro-priate conclusions. In a few cases concepts re-garded as highly demanding were postponed butfor the most part the initial S.M.S.G. 7th grade

15

content was preserved. Abstract principles arecarefully introduced by means of specific concreteexamples.

The Introduction to Secondary School Mathe-maticsVolume I has 4 parts. The first 3 closelyapproximate Mathematics for Junior High SchoolVolume I. Part 4 is a selection of chapters fromMathematics for Junior High SchoolVolume H.The selection was made so that the emphasis ison preparation for algebra. Some topics in geom-etry usually associated with the 8th grade werenecessarily and reluctantly omitted.

It is hoped that the Introduction to SecondarySchool Mathematics will serve to awaken the in-terest of a large group of junior high school pupilswhose ability to learn mathematics has not beenrecognized and that it will build an understandingof fundamental concepts for pupils whose prog-ress in mathematics has been blocked or ilam-pered because of rote learning in an inappitopriatecurriculum. The title was.ralected with the ex-press intent to permit the use of the material atthe 7th or the 9th grade level.

Introdw ie.,n to Algebra(Preliminary Edition]Student texts (IA 1-4) and 8 volume setTeacher Commentaries ( CIA 1-4) Price $8.00

This SMSG course is designed for generalmathematics students at about the ninth and tenthgrade levels. It covers essentially the same mate-rial as the SMSC First Course in Algebra and itfollows exactly the spirit of that course. ( SeeSMSC Newsletter 8, page 22.) However, the lan-guage is simplified, the illustrations are extended,and the pace is slower, so that the course can bestudied with profit over a period of Di to 2 yearsby some students who are not considered collegecapable. The range of ability best served by thiscourse is being tested in an experimental program.Meanwhile, it is presumed that the range is aboutthe middle 50 percent.

The course is based on structure properties ofthe real number system. Thus a student learns the

16

nature of mathematics and strengthens his alge-braic techniques by relating them to basic prin-ciples. The reading material and problems aredesigned to help the student discover ideas. Thenumber line and simpler language of sets are use-ful in expressing these ideas. The material cov-ered includes the usual topics of a first course inalgebra.

Mathematics for High SchoolGeometry with Cewdinates

(Preliminary Edition] 6 volumeStudent texts (GW 1-3) setTeacher Commentaries (C.GW 1-3) Price $8.00

This textboc2 is not a revision of the first SMSGgeometry text, Geometry (revised edition) whichwas described in SMSG Newsletter No. 6, March

IrlloweverTtharare-similariffsome-respects

I. The basic terminology and symbolism isthe same.

2. Solid geometry is integrated with theplane, but long detailed deductive proofsin solid geometry have been eliminated.

3. Real numbers are used freely throughoutthe text, both in the theory and in theproblems.

4. Accuracy in the statement and use ofpostulates, definitions and theorems is em-phasized.

5. Geometric figures are considered as sets ofpoints.

S. GW gives a synthetic treatment of theusual topics of congruence, similarity,parallelism, perpendicularity, area circles,spheres, etc.

Geometry With Coordinates does differ fromthe other SMSG geometry. It presents coordinategeometry as an integral part of the ceurre as rec-ommended by the Commission on Mathematicsof the College Eniranoe Examination Board. Co-ordinates on a line are introduced in Chapter 3,

coordinates in a plane in Chapter 8, and coordi-nates in space in Chapter 9. So coordinate as wellas synthetic methods are used in proving theoremsand solving original exercises starting with Chap-ter 3. Coordinate geometry in the plane is builtin as an integral part of GW, but it was the lastchapter in the other SMSG geometry. There isalso a chapter on vectors. In the chapter on meas-urement of distance, emphasis is placed on theunit of measure.

GW is a deductive geometry using a completeset of postulates and definitions, but the set ofpostulates is not a minimal set. The basic postu-lates, definitions and theorems are motivated byhaving the students do certain experimental exer-cise's and also by appealing to the students' intui-tion. It is hoped that this will make for betterunderstanding of the postulates, definitions andtheorems before the formal proofs of the theoremsare given.

Briefly, Geometry With Coordinates is an in-'tegrated plane and solid geometry, including achapter on vectors; with coordinate systems on aline, in a plane, and in space introduced as earlyas seems practical, so that both synthetic and co-ordinate methods may be used in the proofs oftheorems and originals.


The next three volumes represent a continua-tion of the SMSG program of providing back-ground material for teachers in this case forteachers of the upper grades of the elementaryschool.

Such teachers are included in what is termedLevel 1 in the "Recommendations of the Mathe-matical Association of America for the Trainingof Teachers of Mathematics."

These three volumes have been prepared in thebelief that teachers at any level should not onlythoroughly understand the mathematics theyteach but should also have a good understanding

of the basic concepts in the courses which theirstudents will move on to Accordingly, topics dis-cussed in these three volumes are, for the mostpark those normally met by students in grades7. through 9.

VOLUME VI, NUMBER SYSTEMS

[Preliminary Edition] Privy $2.40A study of the structure of the number systems

encountered in the elementary gradesthe wholenumbers, the non-negative rationah, and integers.A preliminary discussion of real numbers is alsoincluded.

Chapter Headings

1. What is Mathematics

2. Numeration

3. Whole Numbers

4. Rational Number System

5. Coordinates and Equations

6. Real Numbers

Appendix'. Mathematical Systems

VOLUME VII, INTUITIVE GEOMETRY Price $1.25

[Preliminary Edition]A companion volume to. Number. Systems in-

tended to help elementary teachers develop suffi-cient. subject matter competence in the mathe-matics of the elementary school program. Both theapplications_ of number in geometry ( measure-

"ment) and the relationships between geometricelements independent of number are presentedtp help form a foundation for the later study ofgeometry.

Chapter Headings

1. Non-Metric Geometry I

2. Measurement

& Parallelograms and Triangles

4, Constructions and Congruent Triangles

5. Similar Triangles and Variation

6, Non-Metric Geometry sII

7. Volumes and Surface Areas

8. Circles and Spheres

9. Relative Error

VOLUME VIII,CONCEPTS OF ALGEBRA Price $2.40

[Preliminary Edition]Concepts of Algebra provides an understanding

of concepts that give elementary school mathe-matics the very necessary relationship to the totalfield of mathematics and particularly to subse-quent mathematics in the secondary school.

In addition to assistance with terminology, thebook has numerous exercises with a related an-swer section to enable the reader to test his under-standing. These factori make it quite usable byindividuals wishing to strengthen their back-ground for elementary school mathematics. Thefollowing table of contents states the conceptsemphasized.

Part1. Numerals and Variables

2. Open Sentences and English Sentences

3. The Real Numbers

4. Properties of Order

5. Additive and Multiplicative Inverses

DISTRIBUTION OF. SMSG PUBLICATIONSThere is only one source for each of theSMSG publications that may be ordered.The following three sections, I, II and III,relate the publications and the source fromwhich they may be ordered. Care in di-recting orders to tbe correct location willavoid delay and disappointment.

:'Sample Ttuithooki foi.-Giadei 7=12

( Mathematics for Junior High SchoolVol-'Ames I, and II; First Course in Algebra; Geom-etry; Intermediate Mathematics; ElementaryFunctions and Introduction to Matrix Algebra. ) The Yale University Press will continueto publish these, and they will send an an-nouncement concerning prices and improveddistribrition prooedures for the next academie.year to the entire mailing list for this News-letter.

Yale University Press92A Yale StationNew Haven, Connecticut

II. New Mathematical library

Distribution of these Monographs to highschools has been taken over by the L. W.Singer Co. High school students and teachersmay 'order these Monographs ( Numbers: Ra-tional and Irrational; What is Calculus About?;An Introduction to Inequalities; Geometric In-equalities; The Contest Problem Book; TheLore of Large Numbers), price 900 each, f.o.b.shipping point, from:

The L. W. Singer Co., Inc.249 West Erie BoulevardSyracuse 2, New York

The trade edition of these monographs willcontinue to be handled by Random House, Inc.

III. Other SMSG Publications

All other SMSG publications (Studies in Math-ematics; Junior High School MathematicsUnits; Supplementary Units; Conference Re-ports; and preliminary versions of sample text-books) will be available only through thenew SNUG distributor:

A. C. Vroman, Inc.367 South Pasadena Avenue -

Pasadena, California

These publications are listed in the OrderForm included in this Newsletter. A. C. Vro-man, Inc. is a school book depository.

If you are not now on our mailing list butwish to recei ve further issues of this NEWS- .

LETTER, please request, by means of apost card, that your name be added to themailing list.

.-#,Ir

.'SCHOOLMATHEMATICSSTUDY CROUP

Newsletter No. 13July 1962

How THE Teams ARE PREPARED, 3-.

OTHER PUBLICATIONS, .THEIR FUNCTION AND

AvAmtau.rry, 5

SMSG PUBLICATIONS LIST, 6

ORDER FORMS

"Preliminary Editions" only, Vroman,

"Revised Editions" only, Woman, 10

"Sample Text Editions". only, Yale, 11

Other Publications, Vroman, 13

New Mathematical Library, Singer, 15

"PERMISSION TO REPRODUCE THIS COPY-

RIGHTED MATERIAL HAS BEEN GRANTED BY

,G. BogleTO ERIC AND ORGANIZATIONS OPERATING .

UNDER AGREEMENTS WITH THE NATIONAL IN-

STITUTE OF EDUCATION. FURTHER REPFIC-

DUCTIOW OUTSIDE THE 'ERIC SYSTEM RE-

WIRES PERMISSION OF THE COPYRIGHT

- - - _ - -

0 1962 by The Board of Trustees of theLeland Stanford Junior UnivessityAll righti reservedPrinted in the United States of AnseriCa

SMSG

Since 1958 the School- Mathematics Study Crouphas concerned itself with the improvement of teach-ing of mathematics in the schools of this country. TheNational Science Foundation has provided substan-tial funds in support of this endeavor. One of theprerequisites for the improvement of teaching of math-ematics is an improved curriculumone which takesaccount of the increasing use of mathematics andscience in technology-and in other areas of knowledge,and at the same time, reflects recent advances inmathematics itself.

One of the first projects undertaken by SMSC wasto enlist a group of outstanding mathematicians andmathematics teachers to prepare a series of textbookswhich would illustrate such an improved curriCulum.It is not intended that these books be regarded asthe only definitive way of presenting good mathe-matics to students at this level. Instead, they shouldbe thought of as samples of the kind of improvedcurriculum that is needed and as sources of sugges-tions for authors of commercial textbooks.

It is the purpose of this Newsletter to make explicitthe procedures used in generating and distributingSMSC materials, and to outline the method of order-ing textbooks and other SMSC publications duringthe 1962-63 academic year.

HOW THE TEXTS ARE PREPARED

SMSG includes college and university mathema-ticians; teachers of mathematics at all levels, expertsin education and psychology, and representatives ofscience and technology. Each summer since 1958some of these individuals have gathered together onwriting teams to produce text materials. In each case,the, material generated by the writing team duringthe summer months was prepared for photo offsetprocess, for use by schools during the following aca-

. demic year, allowing the printer in some cases no

more than a month to transform typed manuscriptinto paper bound books.

Each of the texts passes through three identifiableeditions: "Preliminary Edition" "Revised Edition""Sample Text Edition". Other SMSG publications,their function and availability, are found listed onpage 5 with the appropriate order forms at the endof this SMSG Newsletter. Since all SMSG publicationsare published at cost under support from The NationalScience Foundation, it should be borne in mind thatNO FREE EXAMINATION COPIES can be sentto individuals or to organizations.

"Preliminary Edition" refers to experimental mate-rials written by the summer writing teams for use inSMSG test-teaching centers throughout the country:The "Preliminary Edition" often consists of isolatedunits which are later to be incorporated into a singletext. They tend to be somewhat experimental, at-tempting under controlled conditions to establishwhether the content is appropriate for the grade forwhich it is written. At the moment, the. ProgramedFirst Course in Algebra (Form MC) and (Form CR)and experimental materials for kindergarten throughthird grade are in this stage of development. Thesematerials may be made available for inspection butonly at some later time, to be announced in the SMSGNewsletter.

"Revised Edition" refers to the transition stage fromthe "Preliminary Edition" to the "Sample Text Edi-tion". The "Preliminary Edition" having been testtaught for one or More years, the "Revised Edition"reflects the revision completed during a summer writ -ing session based on. the information on teachabilityfrom classrooms using the "Preliminary Edition:"Orders for the "Revised Edition" in classroom quanti-ties for September, 1962 were required to be sub-mitted by June to A. C. Vroman, Inc., distributor of"Revised Edition" materials. After classroom quanti-ties have been filled, individual orders will be com-pleted, probably around the first of January, 1963.Geometry With Coordinates, Introduction to Secon-dary School Mathematics, Introduction to Algebra,and Mathematics for the Elementary School Grades4 through 6 will be in this revised edition stage avail-able only from A. C. Vroman, Inc. (see order form,page 10).

"Sample Text Edition" refers to the edition after.the "Revised Edition," if no further revision by SMSCseems required. A non-profit press (Yale UniversityPress at the moment) is contracted to publish the"Sample Texts""sample" in the sense that these areexamples of mathematics which SMSG feels can betaught. When a sufficient number of commerciallyavailable textbooks appears on the market incorporat-ing the suggestions and content of the School Mathe-matics Study Group' materials, the SMSG "SampleText Editions" will be allowed to go out of print.Notice of such a decision will always be announcedwell in advance of the date.

The current materials which are available in "Sam-ple Text Edition" include The Mathematics for JuniorHigh School series and Mathematics for High Schoolseries available from Yale University Press (see orderform, page 11).

OTHER PUBLICATIONS, THEIRFUNCTION AND AVAILABILITY

In addition to "Sample Text" materials generatedby the School Mathematics Study Group there areavailable several series of supplementary materials forstudents and teachers. One of these is SMSG NewMathematical Library for students. For teachers andteacher training institutions the SMSG has producedThe SMSG Studies in Mathematics. In addition theSMSG has held a series of orientation conferences onthe teaching of the new materials which have beenwritten, the Conference Reports now being made gen-erally available. Finally, a Newsletter announcing theprogress of the projects, new publications, and activi-ties of the School Mathematics Study Group is avail-able at no cost by writing to:

SMSC Cedar HallStanford UniversityStanford, California

SMSG PUBLICATIONS LIST

PPreliminary Edition (available for individual use un-til supply is exhausted)

RRevised Edition ( available sometime after 1 January1963)

Db-Title and Description' Cast to

MATHEMATICS FOR THE ELEMENTARY SCHOOLGrade 4, Text (E 4-P) and Commentary

(CE 4.1P and CE 4.2P) 3 vol. set . $5.00 Vroman

SELECTED UNITS, GRADE 4 (E4150) $ .75 VromanGrade 4, Text ( E 4 -1R) 3 parts $2.00 VromanGrade 4, Commentary ( CE 4-1R )

3 parts $2.00 Vroman

Grade 5, Text (E 5-P) and Commentary(CE 54P and CE 5.2P) 3 vol. set .. $5.00 Vroman

Grade 5, Text ( E 5-1R ) 3 parts $2.00 VromanGrade 5, Commentary (CE 5-1R )

3 parts $2.00 VromanGrade 6, Text (E 6-P) and Commentary

(CE 6-IP and CE 6-2P) 3 vol.. set $5.00 VromariGrade 6, Text ( E 6-1R) 3 parts $2.00 VromanGrade 6, Commentary ( CE 6-1R )


INTRODUCTION TO SECONDARY SCHOOL MATHE-MATICS (Grade 7.8 modified) Tent (IS I4P) and Com-mentaries (CIS I-4P) 8 vol. set $8.00 Vroman

INTRODUCTION TO SECONDARY SCHOOL MATHEMATICS

Volume I, Text (ISI-1R ) 2 pads $2.00 VromanVolume I, Commentary (CISI-1R)

2 parts $2.00 VromanVolume II, Text ( ISIHR ) 2 parts $2.00 Vroman-Volume II, Commentary ( CISII-1R )


MATHEMATICS FOR JUNIOR HIGH SCHOOLVolume I, Text (Grade 7) 2 parts . $3.00 YaleVolume I, Commentary, 2 parts $3.00 YaleVolume II, Text (Grade 8) 2 parts $3.00 YaleVolume II, Commentary, 2 parts $3.00 Yale

JUNIOR HIGH SCHOOL MATHEMATICS UNITS1. Number Systems, Text ( U-1 ) $ .70 Vroman1. Number Systems, Commentary

(CU-1) $ .70 Vroman2. Geometry, Text ( U-2) $ .60 Vroman2. Geometry, Commentary ( CU-2) . $ .60 Vroman

Title and Description

3. Applications, Text (U-3)3. Applications, Commentary (CU-3)PKG. Complete Set (one of each of8 above)

SUPPLEMENTARY UNITSJr. H.S. Stipp'. Unit, Text ( JSU)Jr. H.S. Suppl. Unit, Commentary

(CJSU)Essays on Number Theory I. (HSU-1)Essays on Number Theory II (HSU-2)Development of the Real Number

System ( HSU-3)

FIRST COURSE IN ALGEBRAText, 2 parts

FIRST COURSE IN ALGEBRACommentary, 2 parts

INTRODUCTION TO ALGEBRA.Text (IA I-4P) and Commentary(CIA 1-4P) 8 vol. set

INTRODUCTION, TO ALGEBRAText (1A -1R) 4 partsCommentary (CIA-1R) 4 parts

GEOMETRY, Text, 2 parts

GEOMETRY, Commentary, 2 parts

011-

Cost tributer

$ .40 Vroman$ '.40 Vroman

$2.95

GEOMETRY WITH COORDINATESText (GW.1.3P) and Commentary(CGW 1.3P) 6 voL set

GEOMETRY WITH COORDINATESText (GW-1R ) 3' partsCommentary ( CGW-1B ) 3 parts, ....

INTERMEDIATE MATHEMATICSText, 2 parts

INTERMEDIATE MATHEMATICSCommentary, 2 parts

ELEMENTARY FUNCTIONS, Text

ELEMENTARY FUNCTIONS,Commentary

INTRODUCTION TO MATRIXALGEBRA, Text

INTRODUCTION TO MATRIX.ALGEBRA, Commentary

Vroman

$ .80 Vroman

$ .65 Vroman.$ .30 Vroman$ .50 Vroman

Vroman90

$3.00 Yale

$3.00 Yale

$8.00 Vroman

$2.50 Vroman$2.50 Vroman

$3.00 Yale

$3.00 Yale

$6.00 Vroman

$4.00 Vroman$4.00 Vroman

$3.00 Yale

$3.00 Yale

$2.00 Yale

DlsMk sad Don* Ise Cod !OdorNEW MATHEMATICAL LIBRARY

NumbersRational and Irrationalby I. Niven $ 90 Singer

What is Calculus About?by W. W. Sawyer $ .90 Singer

An Introduction to Inequalitiesby E. Beckenbach and R. Beaman $ .90 Singer

Geometric Inequalitiesby N. D. Kazarinoff $ .90 Singer --

The Contest Problem Bookby C. T. Salkind $ .90 Singer

The Lore of Large Numbersby P. J. Davis $ .90 Singer

Uses of Infinityby L. Zippin $ .90 Singer

Geometric Transformationsby. Yaglom-Shields $ .90 Singer


Euclidean Geometry Based on Rulerand Protractor Axioms (SM-2) $ .90 Vroman

Structure of Elementary Algebra(SM-3) $1.40 Vroman

Geometry (SM-4) $2.75 Vroman*

Concepts of Informal Geometry(SM-5) $1.45 Vroman

Number Systems (SM-6) $2.40 Vroman

Intuitive Geometry (SM-7) $1.25 Vroraan

Concepts of Algebra (SM-8) $2.40 Vroman

CONFERENCE REPORTS

Report of a Conference on ElementarySchool Mathematics (CR-I ) $ .40 Vroman

Report on an Orientation Conferencefor SMSG Experimental Centers(CR-2) $1.00 Vroman

Report of an Orientation Conferencefor SMSG Elementary SchoolExperimental Centers (CR-3) $1.00 Vroman

Report for an Orientation Conferencefor SMSG Geometry With Co-ordinates (CR4) $ .40 Vroman

SMSG ORDER FORM"PRELIMINARY EDITIONS" ONLY

A. C. Vromsn, Inc.367 South Pasadena Ave.Pasadena, California

Ihmatity Cost Total

MATHEMATICS FC THE ELEMENTARY SCHOOL

Grade 4, Text (E 4P) and CommentaryICE 4-1P and CE 4-2P) 3 vol. set . $5.00

Grade 5, Text (E 5-P) and Commentary(CE 5-1P and CE 5-2P) 3 val. set . $5.00

CFade 6, Text (E 6.P) and CommentaryME 6.1P and CE 2P) 3 vol. sot. $5.00

_ Selected Units, Grade 4, (E 4150)....$ .75


Texts (IS 1.4P) and Commentaries.(CIS 1-4P) 8 vol. set $8.00

INTRODUCTION TO ALGEBRA

Texts (IA 14P) and Commentaries(CIA 1-4P) 8 vol. set... $8.00

GEOMETRY WITH COORDINATES

Texts (GW 1-3P) and Commentaries(CGW 1.3P) 6 vol. set $6.00

Orders for less than $10.00 value accompanied by remittance willbe shipped at list prices postpaid.

Orders for $10.00 or more value FROM ACCREDITED SCHOOLS willbe allowed 10% discount but transportation will be added.

ORDERS BY INDIVIDUALS must be accompanied by full remittance.Allow 4% sales tax for orders within California.

Bill to

Ship to:

SMSG. ORDER FORM"REVISED EDITIONS" ONLY'

A. C. Vrornan, Inc.367 South Pasadena Ave.Pasadena, California

Graatity Cost TIMMATHEMATICS FOR THE ELEMENTARY SCHOOL

Grade 4, Text (E 44R) 3 parts $2.00

Grade 4, Commentary (CE 44R) 3parts $2.00

Grade 5, Text (E 54R) 3 parts $2.00

Grade 5, Commentary (CE 5IR) 3 parts $2.00.

Grade 6, Text (E 611) 3 parts $2.00

Grade 6, Commentary (CE &4R) 3 parts $2.00


Volume I, Text (ISIR) 2 parts $2.00

Volume I, Commentary (CISI-IR)2 parts $2.00

-Volume II, Text (ISIIIR) 2 parts $2.00

Volume II, Commentary (CISI 14 R)2 parts $2.00

INTRODUCTION TO ALGEBRA

Text (IAR) 4 parts $2.50

Commentary (CIA-111)4 parts $2.50

GEOMETRY.WITH COORDINATES

Text (GWIR) 3 Parts $4.00

Commentary (CGW4R) 3 parts $4.00

Orders for less than $16.00 value accoMpanied by remittance wiNbe shipped at list prices postpaid;


ORDERS BY INDIVIDUALS must be accompanied by full remittanci.Allow 4% sales tax for orders within California.

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10,

SMSG ORDER roftm"SAMPLE TEXT EDITIONS" ONLY

Yale University Press92A Yale StationNew Haven, Conn.

Cost Total

Math for Junior High SchoolVolume I, Text, 2 parts $3.00

Math for Junior High SchoolVolume I, Commentary, 2 parts.. $3.00

Math for Junior High School,_Volume II, Text, 2 parts $3.00

Math for Junior High School,Volume II, Commentary, 2 parts. $3.00

First Course in Algebra, Text, 2 parts $3.00

First Course in Algebra,Commentary, 2 parts $3.00

Geometry, Text, 2 parts. $3.00

Geometry, Commentary, 2 parts.... $3.00

intermediate Mathematics, Text,2 parts $3.00

Intermediate Mathematics,Commentary, 2 parts $3.00

Elementary Functions, Text $2.00

Elementary Functions, Commentary. $2.00

Introduction to Merin Algebra, Text. $2.00

Introduction to Matrix Algebra,Commentary $2.00

Bill to:

Ship to:

See snot silo tortotormatin on Ordering Sample Tat Edition.

INFORMATION ON ORDERINGSAMPLE TEXT EDITIONS

All orders and/or correspondence should be addressed to:

YALE UNIVERSITY PRESS

School Mathematics Study Group Account

92A Yale Station

New Haven, Connecticut.

1. Texts and Commentaries must be listed separately on your or-

der form.

2. Orders will net be acknowledged unless specifically requested

and an acknowledgement form is provided.

3. An educational discount of 30% will be given to schools; acourtesy discount of 10% will be given to libraries, teachers,and other persons directly associated with the field of mathe-

matics.

4. All regular shipping charges by book wit or by truck will bepaid by Yale University Press. If you request any special ship-

ping services (airmail, special delivery, etc.) the extra charges

will be billed on your invoice.

6. Invokes will be sent when the books are shipped unless It >;_is specifically requested that they be sent at a different time..

Please read your invoice carefully. It will supply important in-

formation regarding method of shipment, date books wereshipped, and the number of cartons in each shipment

7.: Place your order well in advance of your needs. Orders will be

filled on a first cow tint served basis.

& Returns may be made without prior permission, subject to the

following conditions: All returns must be sent back in salablecondition, shipping charges prepaid. Our invoice number or

numbers covering the purchase must accompany the shipment

preferably in an envelope attached to the outside of your par!

col. (Your charge-beck memo or other documents should be in

the same-envelope.)

-,SMSO ORDER FORMFORMATIONS

A. C. Woman, Inc.367 South Pasadena Ave.Pasadena, California

Weal Cast Total


Euclidean Geometry Based on Rulerand Protractor Axioms (SM-2). 1 90

Structure of Elemeritary Algebra(SM-3) $1.40

Geometry (SM-4) . $2.75

Concepts of Informal Geometry(91-5) $1.45

Number Systems ISM-6) $2.40

Intuitive Geometry (SM7) $1.25

Concepts of Algebra ISM.8) $2.40

.CONFERENCE REPORTS

Elementary School Mathematics(CR-1) $ .40

SMSG Experimental Centers (CR-2).. $1.00

SMSG Elementary Centers (CD3)... $1.00

SMSG Geometry with Coordinate's(CR4) $ .40

JUNIOR HIGH SCHOOL MATHEMATICS UNITS

1. Number Systems, text (U-1) $ .70

1. Number Systems, Commentary(CU-1) $ .70

....____ 2: Geometry, text (12) .60

2. Geometry; Commentary (CU2)... $ .60.

3. Applications, text (U.3) $ .403. Applications, Commentary (CU-3). $ .40

Pkg. Complete set (ono of each of6 above) $2.95

SUPPLEMENTARY UNITS

Jr. H.S. S.uppl. Unit, text USW $ .80____ Jr. H.S. Sapp'. Unit, Commentary

ICISU) $ .65

Essays on Number Theory I (HSUIi) $ .30

Essays on Number Theory II (160-2) $ .50

Development of the RealNumber System (HSU-3) $ .90

Orders for less than $10.00 value accompanied by remittance will;;be shipped at list prices postpaid.

Orders for $10.00 or more value FROM ACCREDITED SCHOOLS willbe allowed 10% discount but transportation will be added.'.


Bill to:

Ship to:

14

SMSG ORDER FORM'HEW MATHEMATICAL U8RARY"

The LW. Singer Company, Inc.20259 West Erie BoulevardSyracuse 2, New York

Ikaatity Cut Total

Numbers Rational and Irrational $ .90What is Calculus About? $ .90An Introduction to Inequalities .90

Geometric Inequalities $ .90The Contest Problem Book $ 9i)The Lore of Large Numbers $ .90Uses of Infinity $ .90Geometric Transformations $ .90sets of eight titles at $7.20 per set

(All prices are NET f.o.b. shipping point)

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If you are not now on our mailing list butwish to receive further issues of this NEWS-LETTER, plum request, by means of apost card, that your name be added to themailing list.

15

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41


Newsletter No. 24October 1966

"PERMISSION TO REPRODUCE THIS COPT.. I RIGHTED MATERIAL HAS BEEN GRANTED BY

E'.:G:. Beale.TO ERIC AND ORGANIZATIONS

OPERATINGUNDER AGREEMENTS WITH THE NATIONAL IN-STITUTE . OF EDUCATION. FURTHER REPRO.RUCTION OUTSIDE THE ERIC SYSTEM RE.QUIRES PERMISSION OF THE COPYRIGHT

1965 by The Board of Trustees oftheI.thuid &milord Junior University

. An.1Ptiited in the flitted States of Amerki

4

CONTENTS

AVAILABILITY OF SMSG TEXTS, 4

USE OF. SMSG MATERIALS, 6

PRELIMINARY REPORT OF A NEWCURRICULUM PROJECT, 7

ZIP. CODES, 12

REPORTS, 13-20.

REPORT No. 1THE SMSG PROGRAMED LEARNINGPROJECT, 13

ZIP CODE CARD, CENTERFOLD

REPORT No. 2THE SPECIAL CURRICULUM PROJECT, 20

NEW PUBLICATIONS, 28

ORDER FORM, 29

AVAILABILITY OF SMSG TEXTSWhen the first SMSG texts appeared, two ques-

tions arose. The first was how long the textsshould ve kept available, and the second was whatuse the authors of commercial texts should beallowed to make of materials in the SMSG texts.Both these questions were. discussed periodicallyin the SMSG Advisory Board, and a variety of pos-sible policies were care fully examined.

Final resolution of both questions was reached ,

during the past academic year. Reproduced belowis the letter by means of which these decisions werecommunicated to the textbook publishing indus-try.

Dr. Austin J. McCaffrey February 25, 1966Executive DirectorAmerican Textbook Publishers Institute432 Park Avenue SouthNew York, New York 10016

Dear Dr. McCaffrey:The purpose of this letter is to transmit through you

to the members of the American Textbook Publishers Insti-tute current plans for the continued printing of the SMSGtextbooks and our current policy on use of materials in thesetextbooks.

As you know, these textbooks were written becausethe mathematicians and classroom teachers who constitute

41SNISG felt that the mathematics textbooks available a dec-ade ago were not presenting the kind of mathematics ourstudents would need in the society they would find them-selves in after finishing school. The group also felt thatrecommendations and exhortations would be far less effec-tive than the existence of a set of sample textbooks illustrat-ing our point of view and proven feasible in the classroom.

Once these textbooks were prepared, there proved tobe a large demand for them. The group, therefore, decided,to keep them available as long as they were needed, Le.,until the appearance through normal commercial channelsof enough textbooks doing the same job.

In order to make sure that our texts were purchasedsolely because of the content and to avoid any appearance ofcompetition with members of your Institute, we deliberatelystopped short of a letterpress edition. All of our texts areproduced by photo-offset from a typewritten manuscript.We have avoided the use of color. The texts are availableonly in paper covers and so have an estimated lifetime ofless than two years and an estimated cost per student peryear somewhat greater than that of the average hard covertext.

From time to time the SMSG Advisory Board hasasked whether it is necessary to keep the SMSG texts inprint. In order to assist the Advisory Board in its discussions,various kinds of information have been collected. Two yearsago, for example, we sent questionnaires to a random sam-ple of all schools that had purchased SMSG texts at anytime since they first became available. Inspection of thereturns showed, first, that a fairly substantial number ofschools were switching to another text after two or threeyears use of an SMSG text and, second, in almost every suchcase the replacement text was one of the "modern" texts

then becoming available. It may be that we are flatteringourselves, but we conclude that the SMSG texts, perhapsbecause of their extensive teachers' commentaries, providean easy way for a school to move from a traditional programto a modern one.

We have also consulted extensively with state and citySupervisors of mathematics. From them we learned that anon-negligible number of school systems have made heavyinvestments in inservice trainingof their teachers in orderto use SMSG texts. These school systems have urgentlyrequested that the SMSG texts be kept available longenough for them to get a reasonable return from thisinvestment.

We also watch closely the year-by-year sales figuresfor each text. The general pattern has been an increase insales each year until a text has been available for threeyears. At this point a steady decrease in yearly sales begins,and in fact, the decrease for the current school year wasquite marked.

During each of the past two years we have reviewedvery carefully a sample of the newly available commercialtextbooks at the secondary school level. These reviews leaveus with mixed feelings. On the one hand, we are extremelydelighted to see how great the improvement in mathematicstextbooks has been over the last decade. At every gradelevel them is now a variety of texts available with a soundermathematical content and a more modern mathematicalspirit than could be found in traditional texts. We onlyhope that it is proper for us to assume some of the creditfor this improvement.

On the other hand, we were disappointed to see thatmost of these new texts have moved only part way from thetraditional program toward what we consider feasible andappropriate -.or today's school's and students. We had hopedthat by now many authors would have taken an SMSG textas a starting point and would have adapted, improved, andpolished it into a form suitable for commercial publication.

The SMSG Advisory Board felt that this might bedue to a too restrictive policy on use of SMSG materialsor perhaps a misinterpretation of the policy as being toorestrictive. Accordingly, this policy has now been revised asmuch as possible, consistent with our policy of insisting thateveryone have access to these materials and that no one begiven exclusive rights to them. Please note in particularthat the.possibility of a return of part of an author's royal-ties is no longer contemplated. A copy of this new, policy isattached to this letter.

At the same time that this change in policy wasdecided, the SMSG Advisory Board came to a decision onthe availability of SMSG texts. Since, on the one hand, anincreasing number of improved textbooks are available andthe sale of SMSG texts is decreasing but, on the other hand,.the SMSG texts are still useful in helping teachers bridgethe gap between a traditional program and a modem one,it has been decided that these texts will be kept in-print aslong as there is any demand for them and will be allowedto die a natural death when the demand disappears. Thetexts will be kept in their present format, and no revisionsor editorial improvements will be made.

I trust that you will convey this information to themembers of the American Textbook Publishers Instituteand that at the same time, you will convey to them ourappreciation of the good work they are doing and our hopethat they will continue their efforts.

Yours very truly,EGB/san E. G. BegleEnclosure

Use of SMSG MaterialsThe primary purpose of SMSG is the improve-

ment of mathematics in the schools. Therefore,the wide propagation of the ideas in SMSG mate-rials, especially when thoughtfully reworked 'bycompetent authors, is clearly desirable. We en-courage the free and widespread use of these ideasin the production of new textbooks. However,because the materials were produced with publicfunds, the use of SMSG materials should be super-vised.

To implement these principles:(1) SMSG continues to urge authors to improve,

adapt, expand on and draw from materialsprepared by SMSG.

(2) Permission to make verbatim use of currentand projected SMSG materials must be se-cured from the Director of SMSG.

(3) Permission to use SMSG materials will begranted by the Director except in unusualcircumstances.* Publications incorporatingSMSG materials must include both an ac-knowledgment of the SMSG copyright (YaleUniversity or Stanford University, as thecase may be) and a disclaimer of SMSGendorsement. Exclusive license shall not begranted save in exceptional circumstances,and then only by specific action of the Ad-visory Board.

(4) In determining whether an author mayuse SMSG materials, no account will betaken of the author's connections with.SMSG.

(5) In no case will permission to use SMSGmaterials be given until the SMSG publica-tion has been generally available for twofull years.

(6) In difficult cases, precedence should be givento the advantages of propagating the mate-rial.

There are two exceptions. Permission to use extracts froma monograph in the New Mathematical Library must be

secured-from-the-authorT-Arpresent7-ModencteatnitigAids has exclusive distribution rights for the filmed coursefor elementary school teachers.

6

PRELIMINARY REPORT ON ANEW CURRICULUM PROJECT

Since its beginning in 1958, SMSG has devotedthe major part of its efforts to the preparation ofsample textbooks. Of course many other activitieswere also undertaken preparation of materialsfor teachers and for abler students, evaluations ofvarious sorts, etc. but all these together representless time and manpower than the textbook effort.

This aspect of SMSG well is finished. With therevision of a computer text during the summer of1966, a complete set of texts is now available cover-ing the entire range from ",flergarten to the endof high school.

Anticipating this, the SMSG Advisory Boardhas for the last few years been debating whatfurther activities in curriculum development, ifany, should be undertaken. A suggestion to reviseand integrate the present series of texts, eliminat-ing the redundancies and discontinuities resultingfrom the top-to-bottom order in which they wereproduced, was quickly rejected. The Board felt

_ that such revisions and improvements could and_ _should be undertaken by individual authors orteams of authors for commercial publication andthat a single authoritative treatment by SMSGmight well be stultifying.

However, the Board is convinced that curricu-lum- development -must wntinue-indefinitely-fora number of reasons. Most important of these isthat changes in our society will continue to putnew demands on the educational process and re-search and development will be needed to respondto these demands. For example, it has becomesclear only recently that statistics plays such animportant role in our society that every studentshould be given a chance to understand the basicideas of probability on which statistics is based.Similarly, the last decade has seen the use of elec-tronic computers spread so rapidly and so widelythat every student should be given the chance tolearn enough of the relevant mathematics to beable to understand what computers can do andwhat they cannot do.

Another kind of change to which the schoolmathematics program should respond results fromchanges in mathematics curricula which have al-ready been made. Changes in college mathematicscourses seem to be resulting in greater mathemati-cal understanding by those now entering theteaching profession. The widespread use of SMSGand similar texts during the past decade seems tohave contributed similarly to an increased coin-

7

petence on the part of practicing teachers to han-dle new materials. Insery ice training of teachersto prepare them for teaching SMSG has been amajor factor in developing this increased com-petence. Changes Li the elementary school curricu-lum seem to be providing students beginning theseventh grade with a background in mathematicsdifferent from what was the case ten years ago.Changes such as these may well make possiblefurther changes in the secondary school program.

iMost of the new high school texts, includingthose prepared by SMSG, still compartmentalizemathematics in the same way as was done in thepast, e.g., ninth grade algebra, tenth grade geom-etry, etc. It is not unlikely that a different andmore logical sequencing of topics might lead toa more efficient curriculum and one which wouldmake possible the introduction of desirable newtopics.

Curriculum research and development alongthe lines suggested by the above will undoubtedlyrequire a large group effort and can best be carriedout by an organization, such as SMSG, which canbring together the needed variety of teachers,mathematicians, educators, scientists, etc.

For all these reasons, the Advisory Board ofSMSG has decided to take a second look at themathematics curriculum for grades seven throughtwelve.

During the-course-of the which ied-co----this decision, two points were emphasized by theBoard:

1. SMSG is not committing itself to the rapidproduction of a new series of texts for grades seventhrough twelve. Instead the emphasis in this newproject will be on research and development. Thequestion of implementation of the findings of, theproject will not be considered until later.

2. A vigorous attack will be made on the prob-lem of making clear to students the relevance ofmathematics to problems of the real world. It ishoped that the curriculum to be developed willprovide students with a clear understanding bothof the nature of mathematical applications andalso of the great variety of ways in which mathe-matics can be useful in our society.

Letters to SCIENCE and to the MATHEMAT-ICS TEACHER early in 1966 announced thisnew project and urged all those interested in whatmathematics is taught in our schools to submitsuggestions and comments. Several hundredthoughtful letters were received and are beingstudied by those presently involved in the project.

The project is still in a preliminary, formative

8

stage. A small steering committee met briefly lastfall (1965) to provide some direction for a panelthat met late in the winter (March, 1966) in NewOrleans. The panel consisted of university mathe-maticians, applied mathematicians:and practic-ing secondary school mathematics teachers.

Two main guiding maxims emerged from theirdiscussions:

1. The initial segment of the secondary schoolmathematics curriculum should be devoted tothose mathematical concepts which all citizensshould know in order to function satisfactorily inour rapidly expanding technological society. Itwas felt that capable students would be able tocomplete the study of this mathematics in threeyears or less, while the less able students mightprofitably spend four, five, or even six years com-pleting the sequence.

2. The exposition of this mathematics for theaverage to slow-moving student will need to besatisfactorily developed if the project is to be asuccess.

With these maxims:in mind the panel preparedin broad outline a lilt of topics which might con-stitute the mathematics program for grades seven,eight, and nine for capable students and also somesuggestions of later topics.

It is hoped that many students finishing the pro-posed sequential seven-nine program in three years

or-less- will -continue-in grades ten fotweVe arid--will be able to study calculus as well as to under-take some work in linear algebra, vectors, prob-ability, statistics, computers, matrices, and manyother topics not normally found in today's seven-twelve program. It is quite clear that the proposedprogram will make it much easier for most stu-dents interested in careers in engineering andscientific fields to complete the present AdvancedPlacement Program in calculus and also to havesome work in the other topics mentioned above.

The deveiopment of such a program for the lastthree years of high school poses some uniqueproblems because of the diversity of the goals and. interests of the group of students involved at thislevel. Some students will take only one more yearof mathematics after the ninth grade, while somewill want to be mvo ved in studying mathematicsfor two or three more years. It appears that thegroup studying mathematics for three more yearswill essentially divide into two groups. One groupwould be those technically oriented students forwhom the calculus will become the standardtwelfth year course; the second group would con-sist of those equally capable students for whom

9

additional work in probability and statistical in-ference, computers and computing, linear algebra,matrix algebra, etc., will possibly be more profit-able at this time. The deliberations definitely in-dicated that more than one pathway will be avail-able for capable students with different goals andinterests.

During the summer (1966) about twenty prac-ticing secondary school teachers and mathemati-cians met to begin to write detailed outlines ofproposed materials for grades seven through nine.This outlining group adopted some broad generalguidelines:

1. It is assumed that s-tudents entering theseventh grade will have had a modern program atleast in the last few years of elementary school.

2. The mathematics to be introduced in theseventh, eighth, and ninth grades should not onlybe of value to all students as necessary for anyintelligent, responsible future citizen regardlessof occupation but should also provide a properfoundation for futher study of mathematics.

3. The work of other groups who have given--considerable thought to the mathematics suitable -

for these grade levels should not be ignored. Itdoes not seem reasonable to retrace the delibera-tion of the Cambridge Conference, the New Or-

Jeans planning group, etc.; instead we shouldrespond responsibly to their recommendations.

4: New ideas-and-concepts-are-to be refined andgeneralized in a spiral manner over the three-yearsequence.

5. Those aspects of current programs whichhave been successful in providing students withunderstanding and appreciation of the spirit andstructure of mathematics should be retained, butat the same time the new program should dealrealistically with meaningful applications ofmathematics in many different fields.

6. Current mathematical developments thathave arisen as major forces in the many new ap-plications of mathematics, as well as in the devel-opment of mathematics itself, should be broughtinto the curriculum.

Some of the features of the seven-nine sequen-tial program as it is now conceived by the out-lining group are:

1. An attempt is made to fuse arithmetic, alge-bra, geometry in such a way that each one helpsin supporting the development of the other when-ever possible.

2. Geometry is presented in a concrete, intui-tive, descriptive way. When a new concept isintroduced, the focus is on its essential features.

10

-Its more complex aspects and its relation to otherConcepts are treated late). The treatment of ge-ometry fosters uoderstanding of the concrete basisand the intuitive significance of geometric ideasbefore they arc studied on a rather formal deduc-tive basis.

3: One, two, and three dimensions will be con-sidered in many geometrical questions whereapplicable. Coordinate geometry will appear asappropriate in helping describe sets of points alge-braically. Solution sets of algebraic equations willbe interpreted or described geometrically.

4. The .process of model building in appliedmathematics is the construction of a mathematicalmodel that will help one to better understand asituation that occurs in a complex environment,since the analysis of such a mathematical situationoften enables one to learn more about the originalphysical situation. This process will be developedat appropriate places in the seven-nine sequence.

5. Sonic relaxation in the present stress onstructure' may be noticeable, but structure is stilldefinitely one of the unifying themes.

6; Short deductive sequences will appear in--grades seven through nine using properties dis-covered in the intuitive development of both alge-bra and geometry. Some problems from numbertheory will be used to illustrate various methodsof proof.

7. Appropriate materials will be included sothat the student will have some appreciation forthe mathematical activities associated with com-puter use and the role of computer in our tech-nological society. Mathematical algorithms ex-pressed in flow chart notation will offer an intro-duction to computer activity and require thestudent to have sufficient understanding of themathematics for practical exploitation.

8. The concept of function will be consideredearly and will be used in many different types ofmathematical content whenever appropriate.

9. The concept of a vector (displacement) ap-pears in grades eight and nine.

10. Probability appears in grades seven andeight with some statistics in grade nine.

----117-Nunieration-systenis will vet little treatmentbecause it is assumed this topic will have beencovered in elementary schoei

12. The set concept and set notation will beused 'when convenient, but it will not be over-played. It is assumed that the students have thatconcept from elementary school.

13. Notation and terminology introduced in

11

these grades will be compatible with present-dayusage in mathematics.

During the course of the next few years the ten-tative outlines prepared this summer will be re-fined and extended to grades ten through twelve.Many of the units in these outlines will be testedexperimentally in the classroom. Periodic progressreports of this project will appear in this news-letter.

ZIP CODESStarting January 1, 1967, the U. S. post office

wilt accept fourth class mail only if the zip code isincluded with each address. Since SMSG news-letters arc mailed at the fourth class rate, it isnecessary that the SMSG newsletter mailing listbe updated to include zip codes.

If you wish to continue to receive SMSG news-letters, please fill out the address card in the cen-terfold of this newsletter and return it to SMSGheadquarters.

Naturally, the above applies only to addresseswithin the USA: Those residing outside the USAwill continue to receive SMSG newsletters asusual.

12

REPORTSReports on a number of SMSG projects, studies,

and evaluations have appeared in earlier issues ofthis newsletter.' Some studies, however, have re-sulted in reports too lengthy to be communicatedin this fashion. Consequently, a new series of pub -lications SMSG Reports has been started. Thefirst two numbers in this series are now available.Single copies may be obtained by means of a postcard request to

ReportsSMSG Cedar HallStanford UniversityStanford, California 94305.

Summaries of these first two reports follow.

Report No. 1THE SMSG PROGRAMED LEARNING

PROJECT

In the late fifties, considerable interest in pro-gramed learning was generated from varioussources, notably from a large number of writersand publishers. In response to the interest thatwas rampant at the time, the SMSG Ad Hoc Com-mittee on Programed Learning was formed in1961 to advise the Director of SMSG on proce-

-dures-for-a-careful-study of-programed-learningwith specific reference to SMSG materials. Thefollowing recommendations were made at the ini-tial meeting of the committee which took place atHarvard University on May 2, 1961:

1. Mathematicians from both high school anduniversity levels should be taught to constructprograms.

2. The Director of SMSG should locate a suit-able number of qualified individuals who coulddevote part of their time during the 1961-62 aca-demic year to the project.

3. Each section of the SMSG First Course in Al-gebra should be translated into programed form.

4. A variety of programs should be prepared.5. The program writers should retain the con-

tent and sequence of the SMSG First Course inAlgebra in order to make comparisons possible.

6. The problem of motivation should be keptin mind in writing the program.

7. Feedback should be obtained from studentswho try the program.

8. The objective of the first part of this SMSG

13

study should be to verify the hypothesis that theSMSG materials can be .presented as effectivelyaccording to prescribed criteria, through programsas through conventional classroom- textbookprocedures.

9. Evaluation of any program should involvethe use of tests which require students to solveproblems and recognize concepts outside the text.

Specific remarks spelled out the basic thinkingbehind these general recommendations. For ex-ample, the recommendation to translate the SMSGFirst Course in Algebra was motivated by the de-cision that programing this course would providemore useful information, even though it was feltthat the SMSG 7th grade course would be techni-cally easier to program. Again, for example, thevariety of programs indicated was to include aconstructed response program featuring smallsteps with some branching and a multiple choiceprogram featuring a scrambled book format withexplanations of incorrect answers. Occasional in-clusion of larger steps in the constructed responseprogram and variation in size of the unit in themultiple choice program were recommended forthe purpose of experimentation!

With these recommendations of the preliminarycommittee, the Director appointed a Panel onProgramed Learning consisting of R. C. Buck,

--University- of-Wisconsin;- E.- E.- Hammond, -Jr.,-Philips Andover Academy; L. D. Hawkinson. SanFrancisco Unified School District; J. G. Holland,Harvard University; W. J. McKeachie, Universityof Michigan; E. E. Moise, Harvard University; H.O. Pollak, Bell Telephone Laboratories; and D.W. Taylor, Yale University.

As a guide for the writing teams, a chapter bychapter summary and statement of objectives forthe algebra course was prepared by V. H. Haag,H. 0. Pollak, and C. E. Rickart who were mem-bers of the original writing team on the SMSGFirst Course in Algebra. Preliminary to the fullscale production of materials, a Manual for Pro-gramers was drafted by Leander W Smith, Coordi-nator of the Programed Learning Project. Themanual was intended as a basic document to pro-

-vide-persons-having-a- thorough-knowledge-orthemathematics with information that they wouldneed for the actual construction of programs. Itincluded some background material on the psy-chology, an overview of the state of the art of pro-graming, specific recommendations for the anal-ysis of the content of the SMSG First Course inAlgebra, and. notes on the construction of items.

14

The manual has since been revised and incorpo-rated and published in a more comprehensive re-port of the SMSG Programed Learning Project.

Twelve writing centers were established to workon a part-time basis during the 1961-62 academicyear. Each of the twelve writing centers consistedof three or four mathematicians and/or mathe-matics teachers whose acquaintance with SMSGFirst Course in Algebra stemmed from participa-tion in the original writing team, the evaluationcenters, or teaching SMSG mathematics to teach-ers. Front each of these centers, representativeswere assembled at Yale University for a workshopin August 1961, to acquaint a core of writers withprogramed instructional materials and the pro-cedures for producing them; to prepare sampleunits in constructed response and multiple-choiceformats which could serve as models for the cen-ter writing teams; and to take on writing assign-ments and suggestions for the organization of thetwelve centers.

Participants in the Workshop were: D. W.Blakeslee, San Francisco; M. P. Bridgess, Boston;F. D. Jacobson, New Haven; H. Jones, Stillwater;W. W. Matson, Portland, Oregon; 0. Peterson,Emporia, Kansas; P 0. Redgrave, Norwich, Con-necticut; W. Storer, Des Moines; and H. Swain,Winnetka. The participants formed two writingteams; one to produce a constructed response pro-

-gram; the-other to-produce a-multiple-choice-pro--gram. With the experience of writing as a team inNew Haven, the participants were then able toanticipate problems of producing programs andorienting other members of the centers.

Typical operational procedures for each of thewriting centers consisted of first, internal criticismof a draft by members of the center and sugges-tions for refinement by the center chairman. Withfurther refinement and editing of subsequentdrafts, the revised drafts were sent to SMSG (thenat Stanford University) for ditto reproduction andcriticism by reviewers selected for this purpose.Each reviewer commented on the particular phaseof the work in which he was best fitted: the psy-chologist commented on the programing per se,the mathematician on the correctness of the pres-entation of the mathematics, and the teach-ers onthe appropriateness of language to the high schoolstudent. On receipt of the comments from the re-viewers, the writing team reconsidered the lan-guage, mathematics, and- programing technique,and prepared another draft for lithographing.

By sprang (March 1962),, the lithographed ma-terials were made available to experimental cen-

15

ters. Prior to the use of these materials, a two partpre-test was administered to serve as an inventoryof skills and concepts already in the command ofthe students. Writers were found to generate pro-gramed materials in the summer of 1962 for usein schools in the 1962-63 academic year. This writ-ing group consisted of: John D. Baum, David WBlakeslee, M. Phi !brick Bridgess, Mrs. MarjorieFrench, James E. Gilbert, Arthur A. Hiatt, Ste-phen Hoffman, Mrs. Margaret Ma tchett, WilliamW Matson, Mrs. Persis 0. Redgrave, WinfieldRoberson, Robert E. Schweiker, William Storer,George Truscott, and Mrs. Helen We he. The par-ticipants were subdivided into two writing teams:one to revise the constructed response program,and the other to revise the multiple-choice pro-gram. Mr. Bridgess chaired the multiple-choice(Form MC) team, and Mr. Blakeslee chaired theconstructed response (Form CR) team.

The writers of the multiple-choice programspent considerable time anticipating distractors(alternatives which require careful discriminationand knowledge of skills and/or concepts). Themultiple-choice format enabled some preservationof text but suffered in its ability to elicit compu-tational practice because of the form of presen-tation. The multiple-choice program that was pro-duced that summer, although a scrambled bookin form, was essentially a single-track program

through which a student would proceed. Verylittle effort was made to capitalize on the multipletrack program that might be possible given more,time to write.

The writers of the constructed response pro-gram, on the other hand, found the fragmentationinto small steps of the earlier version quite a prob-lem. There was an initial tendency to use lengthysets of problems for practice, over-cuing, and littlesynthetic material..Finding a need for synthesis,the writers redefined "frame:' A "frame" had beenconceived as a short passage, three to five lines,with a significant word or words missing. Theresponse sought was then to contribute to thelearning of the skill or concept. Each frame wasthen defined as a "bit" or "response elicitingitem:' As the summer progressed, the writers be-gan to group exercises of similar nature into abox or frame; they then sought to use the "frame"as a means of identifying a recognizable step inthe learning process. Each frame evolved as a con-ceptual unit; that is, an identifiable step towardthe mastery of a skill or concept.

The ease of programing seemed to depend onthe explicit structure of the content presented in

16

the text and the independence of topics being de-veloped. Programing of proofs was a challenge, Toovercome part of the difficulty here, use was madeof the constructed panel in the GR format. Inthis, the student was led to complete steps (or tosupply reasons for steps) in a proof, transferringthese responses to appropriate blanks in a Panel tocomplete the proof. The constructed proof wasthen compared with a model proof as it mightappear. Thus, the Panel construction served tointegrate the small pieces of the proof into awhole. Also, in the process of reconstruction, morethan one pass were made in the proof.

In the spring of 1962, a design was planned forexperimentation during the 1962-63 academicyear. The design called for testing the followingfour modes of instruction:

MC the multiple-choice programed text;CR the constructed response programed text;MCR the constructed response machine for-

mat (unbound CR in Koncept-O-GraphKOG-7);

9F standard SMSG First Course in Algebra.Each mode was to be used under certain of the

following conditions:Time-paced with a teacher:

assigned with a due time or due date in blocksor sections with a teacher to control disburse-ment and to answer questions and discuss orexplain any mathematics.

Time-paced with a monitor:assigned with a due time or due date in blocksor sections with a monitor to control disburse-ment and to answer nonmathematical ques-tions.

Self-paced with a teacher: .

in classroom or home with a teacher to recordtime and to answer questions 'and discuss orexplain any mathematics.

Self-paced with a monitor:in classroom or library with a monitor to re-cord time and to answer nonmathematicalquestions.

Since teachers were often adverse to the notionof self-paced classes, almost all self - paced classes

. reverted to some form of time-pacing at some timedariffgth-FTea-f.-Sifithese classes thus consti-

tuted a conglotheration of methods of presenta-tion rather than pure form of self-pacing, theywere eliminated from the analysis. The four treat-ment groups chosen for analysis were then, theMC (multiple-choice), CR (cons; -ucted response),MCR (constructed response, ma.:hine), and 9F'(standard SMSG text) programs ail time-paced

17

at 9th grade. The MCR classes were supervisedby monitors while all others were supervised byteachers.

There was intense negative reaction from someof the teachers and students against the boredomthey found resulting from the programs. Much ofthis reaction appeared to be generated by thesameness of format. In the case of the constructedresponse program, boredom was intensified amongthe faster students who were forced to go throughlong series of items designed primarily to teachthe slower students. In the multiple-choice pro-gram, which provides for branching and skippingto care for the individual differences, there wasobjection to the "page-Ripping" involved in thescrambled book technique; they also objected tobeing unable to review systematicaliy without re-tracing all the steps in the program. In addition,the machines were found to be unsatisfactory. ByJanuary 1963, about half the machines were un-usable because of mechanical deficiencies.

From the widespread dissatisfaction of teachersand students in the 1962-63 classes came the feel-ing that the "pure form" programs had madeSMSG mathematics less palatable than the text onwhich they were based. Hence, in the spring of1963 the Project Coordinator prepared a sample.which combined the features of both earlier ver-sions into a hybrid program. The features whichwere felt essential included:

(1) immediate confirmation on most constructedresponse items;

(2) confirmation and correction of errors onmultiple-choice items without scramblingthe text;

(3) inclusion of conventional textual material;(4) inclusion of reviews and problem sets with

confirmations given in the back of the book;(5) inclusion of optional sections for more able

students and for students who want or :'eedadditional help;

(6) inclusion of index, table of contents, tailor-made response sheets;

(7) variation in format to relieve boredom;(8)-provision _for- skippi ng-i tems-by-thoseTwho

have successfully completed Criterion items.The hybrid program, SMSG Programed First

Course in Algebra (Form H), thus emerged in pre-liminary form after eight weeks of intensive writ-ing in the summer of 1963. This writing teamconsisted of D. Blakeslee, M. P Bridgess, F Elder,J. C. Hammock, F. Jacobson, M. Matchett, W.

18

Matson, L. W Smith, W. Storer, M. Wecheler, andM. Zelinka, chaired by Mrs. P 0. Redgrave. In1963-64, an attempt was made to evaluate Form Hin a large scale experiment with Forms CR, MC,9F, and the SMSG Introduction to Algebra (IA),using common testing on all groups. (The SMSGIntroduction to Algebra covers essentially thesame material as the SMSG First Course in Alge-bra, but featuring slower pacing, more extensiveillustrative materials, and simplified language.)For the evaluation, pre-tests and post-tests were

iadministered. In the analysis, all items from thepost-tests were coded as to the level of intellectualactivity which they measured, Lo-cognitive meas-uring intellectual activity equal to or less thanmanipulating, and Hi-cognitive measuring intel-lectual activity greater than manipulating. .A fol-lowup study was made in May 1964 of about 450students who had been in the 1962-63 classes todetermine whether retention of the algebra theyhad studied was related to the mode they had used.

During the 1964 SMSG Summer Writing Ses-sion, a writing team chaired by D. Blakeslee andconsisting of W. G. Chinn, F Jacobson, M. Match-ett, W. Matson, and P Redgrave comprehensivelyrevised the hybrid text. This revision includedsignificant. changes in the mathematical contentof 9F based on critical reviews by the PLP writersand analytical comments from members of the

SMSGAdvisory Board, notably those of P.D. Lax,New York University, and H. 0. Pollack, BellTelephone Laboratories.' In addition, a TeacherCommentary wasprepared. The final revision ofthe hybrid text concluded SMSG's examinationof programed instruction.

In summary, the results of the 1952-63 studyof the four modes indicated that both the con-structed response (CR) and multiple-choice (MC)programs were significantly better than the stand-ard SMSG text (9F) while there was no significantdifference between CR and MC (all significancewas at least at the .05 level). Also, both CR andMC were significantly better than the machineconstructed response (MCR) while' there was nosignificant difference between MCR and 9E Whilemany of the differences among treatment groups

were-significant, they were small in amount.The results of the 1963-64 study indicated that

with Lo-cognitive post-test measure of perform-ance as the criterion variable, the MC, H, and 9Fprograms were better than.CR, and both H and9F were better than IA. While the difference be-tween MC and IA was not significant, it ap-proached significance. MC, .H, and 9F tended to

19

cluster together in their effectiveness to teach Lo-cognitive skills; likewise, CR and IA ter,.led to beequally effective, though significantly less effectivethan the other three modes. With the Hi-cognitivepost-test measure as the criterion variable, theresults tended to indicate that all of the modeswere equally effective for teaching Hi-cognitiveskills. Only one difference was found to be sig-nificantly different: H > CR. ,

In the follow-up study of 1964, the null hy-pothesis of no difference among treatments wasnot disproved; that is, one year after completingthe course, performance on the follow-up test wasnot significantly different for students who hadused MC, CR, or 9F.

Original literature on programing did not pro-vide for a program "mix:' However, the mixedform tried in the 1964 writing had more clearlymet the criteria of improVed student performanceand greater philosophical satisfaction to the writ-ers: In view of this, an examination of programedmaterials that are feasible and promising shouldinclude such hybridization, for consideration.

Report No. 2

THE SPECIAL CURRICULUM PROJECT:Pilot Program on MathematicsLearning of Culturally DisadvantagedPrimary School Children

IntroductionThe purpose of 'the Special Curriculum Project

is to study the readiness of culturally disadvan-taged children for learning at school entrance andto follow their progress over the beginning schoolyears with the aim of developing more effectivematerials for their learning of mathematics.

This project grew out of the Conference onMathematics Education for Below AverageAchievers sponsored by SMSG in April of 1964.On the basis of papers and discussions at this con-ference and of recommendations of an ad hoccommittee, observations classes at the kindergar-ten and first grade levels were established for the1964.65 school year in six cities.

Fifteen classes, seven at kindergarten and eightat first grade, were selected as experimental classes.These classes were all located in disadvantagedareas of the six cities. All used the existing SMSGK and 1 texts, and, in addition, all were providedwith a variety of manipulable materials for usein mathematics instruction.

20

Two kinds of comparison classes were includedin the study. The first was designated as a socio-economic comparison group and included twokindergartens and two first grades in more ad-vantaged areas of the same metropolitan areaswherein the experimental classes were located.The second comparison group included one kin-dergarten and one first grade class located in thesame city as two of the experimental classes. Thesetwo classes used a curriculum other than SMSG.

The majority of these experimental and com-parison groups were followed through the 1965-66school year; however, the report here summarizeddeals only with individual testing of the childrenduring the 1964-65 school year.

Design and MethodologyThe study was based on the assumption that

children from disadvantaged homes come toschool with fewer experiences which enable aca-demic success than do children from more advan-taged homes. The emphasis of this pilot researchwas empirical,, with major 4ttention focused. onmeasuring:

'1) certain dimensions of readiness for schoollearning at kindergarten and first gradeentrance;

2) certain facets of concept development;3) development of particular mathematical

concepts.With the baseline of test scores at the begin-

ning of the school year, it was then possible tomeasure change in performance over the schoolyear with the combination of the SMSG elemen-tary texts, special consultation assistance for theteachers, and the use of many concrete, manip-ulable materials to aid the children in under-standing mathematical concepts.

The portion of the.study here reported may bediagramed as follows:

School Year September January hine

4- SMSG Experimental Program

(Disadvantaged and Socio-EconomicComparison)

Non-SMSG Curriculum

(Curriculum Comparison)

Initial Midyear FinalInventory Inventory Inventory

Treatment

Evaluation ofTreatment

Effects

21

The individually administered tests, labeledabove as Initial, Midyear, and Final Inventory,were developed to minimize whatever differentialmight exist between the disadvantaged and moreadvantaged children in skills related to test-taking.The concepts and knowledge measured at each ofthe three individual testing sessions are shown inthe following table.

INDIVIDUAL TESTS BY GRADE

Assessinents Made

Recognition .

Objects X XPhotographs X XDrawings X X

Initial Midyear Final

K 1st K 1st K 1st

Vocabulary X

Visual MemoryObjectsPictures

X X X XX X

Color InventoryMatching X XNaming X X X XIdentifying X X X X

Geometric ShapesMatching X XNaming X X X XIdentifying X X X X

Pairing X

Equivalent Sets X X

CountingButtons X X X X XMembers of a set X X X XRote X X X XRote by Tens X

Number SymbolsIdentifying X X X X XNaming XMarking X X X X X

Place ValueNaming XForming X

Ordinal Number X X

Ordering X X X X

Classifying X X X X

22

As an explanation of the assessments listed inthe above table, the Initial Inventory was plannedas an evaluation of readiness of the children tolearn mathematical concepts. This readiness isdependent upon experience and development inmany areas. If the child is to learn to abstract andconceptualize from experience with concrete ma-terials, it is important to ascertain whether he canrecognize, by giving names to, the concrete mate-rials being used (Recognition Objects).. Also,since color is an important classificatory principlein the early school years, it is important to learnwhether the children can perceive and match thesame hues, name the colors, and identify an objectwhose only property differentiating it from theothers in the series is its color (Color Inventory).

Two other facets of cognitive development wereassessed. The first was the ability of the child tomake a transition from recognition of a concrete

.- object to recognition of a photograph of it, andthen to a line drawing of that object. The secondwas one kind of.meeiating response: visual mem-ory. The progression from concrete to conceptualthought, upon which mathematical learning isbased, requires that the child be able to form men-tal representations of objects he has previouslyseen but which are physically absent. Therefore,the visual memory assessment was used.

Assessment of performance on tasks more di.'rectly mathematical in nature included matchingand naming of geometric shapes, counting, recog-nition of numerals, ordering objects by size, andclassifying them by shape and color. A number ofthese assessments made in the initial testing wererepeated or extended in the later testing sessionsto ascertain growth through the school year. Inaddition, other kinds of tasks (Vocabulary, Pair-ing, Equivalent Sets, Place Value, Ordinal Num-ber) were added.

Results and Discussion.

The results are complex to present briefly sincethe study deals with experimental classes plus twokinds of comparison classes at two grade levels. Inaddition, the summary of results by curriculumand socio-economic level must be prefaced by thegeneral finding that there was great variabilitywithin the classrooms of the disadvantaged chil -.dren as well as great variability in performancebetween these classes at the end of the school yearas well as at the beginning.

The results are presented by the specific assess-ments on the criterion of whether significant dif-ferences were found between samples. For this

23

study the probability level accepted for signifi-cance of difference between groups was .01.

1. Initial InventorySignificant differences which were found on the

tests at the beginning of the school year were allin the direction of poorer performance of thedisadvantaged children when compared with themore advantaged as well as the curriculum com-parison group. The following table shows thespecific assessments on which the experimentaldisadvantaged sample performed significantly lesswell.

Assessment

Visual Memory Objects

Color Matching

Color Naming K, 1

Color Identifying K, 1

Counting Objects (Buttons) K, 1

Rote Counting 1

Recognition of Number Symbols{ Ki

Marking Number Symbols 1

Grade ComparisonLevel Group

K Curriculum

K Curriculum

Socio-economic

Socio-economic

Socio-economic

Both

Both

Socio-economic

Both

Both

There were more specific tests at the beginningof the school year (Initial Inventory) on which nodifferences were found. between the performanceof the disadvantaged kindergarten sample and oneor both of the two comparison groups thanof thefirst grade sample. This supports findings fromother studies which show an increasing lag in theperformance of disadvantaged children as theyprogress from grade to grade.

The specific tests on which the disadvantagedkindergarten children showed no difference inperformance when compared to one or bot.;the other two groups were:

Recognition Objects, Photographs,Drawings

Visual Memory ObjectsColor Matching, Naming, Identifying'Rote CountingNumber Symbol Recognition, Marking

For the disadvantaged first grade children, theonly tests on which no differences were obtainedwere:

24

Recognition Photographs, DrawingsColor Matching

H. Midyear InventoryThe tests at the middle of the school year meas-

ured primarily mathematical learning. At thispoint in time the pattern of differences betweenthe groups was quite different from what wasfound at the beginning of the school year.

The experimental, disadvantaged kindergartensample performed significantly better than did thecurriculum comparison sample on GeometricShapes Matching and Naming, while the disad-vantaged first grade children performed signifi-cantly better than the curriculum comparisonchildren on Geometric Shapes Naming, and onPairing. These differences are likely attributableto differences in the curriculum but clearly attestto the ability of the disadvantaged children tolearn the SMSG curriculum.

The performance of the disadvantaged firstgrade children was poorer than that of the socio-economic comparison first graders on Vocabulary,Geometric Shapes Naming, and Number Sym-bol Marking. They performed less well than didthe curriculum comparison sample on numbersymbol recognition only.

There were no tests at midyear on which thedisadvantaged kindergarten children performedat a significantly lower level than either of the twocomparison kindergarten groups. This latter find-ing suggests that 'intervention at kindergartenwith a structured mathematics program may di-minish the discrepancy in mathematics perform-ance between disadvantaged and more advantagedchildren which becomes readily apparent by thirdor fourth grade.

III. Final InventoryBy the end of the school year, the performance

of the disadvantaged kindergarten children wasstatistically no different from either comparisongroup on most of the assessments made. Theirperformance on Visual Memory Pictures was sig-nificantly poorer than that of the curriculumcomparison kindergarten group. On Geometric

I Shapes Naming and Identifying, they continuedto perform better than the curriculum comparisongroup. On Counting Buttons. and on NumberSymbol Identification and Marking, the disadvan-taged kindergarten children performed less wellthan the more advantaged kindergarteners, butthese differences may be considered directionalonly since they reached the .02 and .05 levels of

25

significance, respectively, instead of the accepted.01 level. By comparison to the significant differ-ences between groups found at the beginning ofthe school year on these tests, their progress overthe year was considerable.

For the fir,* grade classes, the Final. Inventoryresults were somewhat different. The one test onwhich the disadvantaged first graders performedsignificantly better than did the socio-economiccomparison group was on Counting Members ofa Set. On all other assessments in which significantdifferences were. found, the performance of thedisadvantaged first grade children was poorer thanthat of one or both of the comparison groups.Those assessments on which the performance ofthe experimental first graders was poorer thanboth comparison groups were:

Rote CountingRote Counting by TensNumber Symbols Naming and MarkingPlace Value Forming

On Ordinal Number, their performance waspoorer than that of the curriculum comparisonclass. .

It is important to emphasize, however, that thedisadvantaged first graders' performance im-proved markedly over the course of the schoolyear so that their performance on several taskswas not significantly different from the two com-parison groups at the end of the school year al-though it had been at the beginning or middleof the year.. This is the case for Number SymbolsRecognition, Geometric Shapes Naming, Color

Naming and Identifying.

Summary and ImplicationsThe report summarized has dealt with individ-

ual test results only, although classroom observa-tions, teacher reports, and a group test adminis-tered at the end of the school year provided otherkinds of data. Apart from the substantive findingsreported, certain trends deserve mention. Thevariability in performance within the disadvan-taged classes as well as between them was foundwith regularity although this intra- and inter-classvariability could not be detailed in this summary.This variability demands a more careful look atthe factors affecting the individual child's per-formance and suggests caution in grouping thefindings of a number of classes on the basis that:they are all composed of children who can bedescribed as disadvantaged.

The changes in performance of the disadvan-

26

taged kindergarten children over the 'year were,on many of the tests, different from those changesobserved in the first grade children. At this stageof the analysis; those changes which are effects ofthe differences in curriculum content at the twolevels cannot be partialled out from those whichmay be attributed to the effects of earlier intervention with a structured mathematics program at thekindergarten grade.

Further analyses needed include intercorrela-tions among the various tests and between per-formance on the individual and group tests.Studies of the kinds of errors made, particularlyon the number concept measures, may provideimportant insights into the process of learningmathematical ideas.

A further research need is for a longitudinalstudy to better evaluate the significance of provid-ing experiences in mathematics at the kindergar-ten level on children's continued progress in theintermediate grades, a time at which the disad-vantaged child's cumulative deficit has, in thepast, become so apparent.

27

NEW PUBLICATIONS

SMSG has prepared units on probability forgrades K through nine. These units are arrangedin four volumes. For the primary grades the majoremphasis is on the 'recognition that some eventsare uncertain. The notion of one event being"more likely" than another is developed. In thevolume designed for the upper elementary grades,the appropriate mathematics of uncertainty isintroduced.

A more formal treatment of probability forjunior high school students is based on the lan-guage and symbolism of sets. Part I is largely intui-tive and is designed for grade seven. Some of themore advanced material of Part II is best studiedafter SQMC experience in algebra. The junior highschool volumes are prepared in hybrid pro-gramed form and thus may be used as self-studyunits.

In all the units experiments and games are usedto motivate mathematical generalizations. For theprimary and intermediate grades, specially de-signed spinners have been prepared for this pur-pose and are available in classroom sets.

See order forna on facing page.

28

MSG ORDER FORMRevised Editions

Available only from A. C. Vroman, Inc.367 South Pasadena AvenuePasadena, California

Probability for Primary GradesStudent Text

Probability for Primary GradesTeacher's Commentary

Cut Tebl

$ .50

$ .75

Probability for Intermediate GradesStudent Text $ .75

Probability for Intermediate GradesTeacher's Commentary $1.00

Classroom set of Spinners forPrimary Grades $5.00

Classroom set of Spinners forIntermediate Grades $5.00

Introduction to ProbabilityPart 1'Basic Concepts, Student Text $1.00

Introduction to ProbabilityPart 2Special Topics, Student Text $1.00

Orders for less than $10.00 value accompanied by remittance willbe shipped at list prices postpaid,



Bill to:

Ude to


Newsletter No. 25

December 1966

ARTICULATION OFCONTENT OF SMSG TEXTS;GRADES 7-10

'PERMISSION TO REPRODUCE THIS COPY-RIGHTED MATERIAL HAS BEEN GRANTED BY

E. G. Beg leTO ERIC AND ORGANIZATIONS OPERATINGUNDER AGREEMENTS WITH THE NATIONAL IN-STITUTE OF EDUCATION. FURTHER REPRO-DUCTION OUTSIDE THE ERIC SYSTEM RE-WIRES PERMISSION OF THE COPYRIGHTOWNER."

0 19116 by The lewd of Trustees of theLeland Stanford Junior UnirenityMI riling wiseMIdPrinted in the United States of America

CONTENTS.Comments on Articulating SMSG Mathematics

for Junior High School with students' prior SMSGexperiences. p. 5

VOLUME 1 PART I CHAPTERS 1-8, 7VOLUME 1 PART II CHAPTERS 9-14, 12VOLUME 2 PART I CHAPTERS 1-6, 15VOLUME 2 PART II CHAPTERS 7-13, 20

Comments on Articulating SMSG First Coursein Algebra with students' prior SMSG expe-

.nences. p..23PART 1 CHAPTERS 1-9, 25PART II CHAPTERS. 10-17, 32

Comments on Articulating SMSG. Geometrywith students' prior SMSG experiences. p. 35

PART I CHAPTERS 1-10, 37PART II CHAPTERS 11-17, 45

The textbooks which SMSG has prepared donot form a closely integrated sequence. In gen-eral, they are written for students whose previoustraining in mathematics was in a traditional pro-gram. Consequently, each text contains some ma-terial which has already been discussed in otherSMSG texts 'for earlier grades.

A number of teachers have asked for advice asto how to proceed, especially in grades seventhrough ten, with students who have had previousSMSG experience.

A committee of classroom teachers who hadhad substantial experience with SMSG texts andwith the problems of articulating them wasaaskedtp prepare suggestions and recommendations.This issue of the SMSG newsletter contains thereport of this committee.

The members of the Articulation Committeewere:

Sarah T Herriot, Palo Alto, California(chairman)

Vincent Brant, Baltimore County, MarylandFlorence Elder, West Hempstead, New YorkTemple Franklin, Arlington, VirginiaMildred Keiffer, Cincinnati, OhioMax Sobel, Montclair, New JerseyRuth Stone; Palo Alto; California

Editorial assistance in the preparation of thereport was provided by:

Sarah Ferguson, San Francisco, CaliforniaMary Huggins, Palo Alto, California

The purpose of the report of this SMSG com-mittee is to help articulate the exploratory, infor-mal work SMSG provides in' the elementary yearswith the more unified and abstract content spiral.ing on through the junior and senior high schoolSMSG mathematics curriculum by offering sug-gestions for fitting prior SMSG experiences intoa systematically enlarging body of knowledge.

It is the committee's consensus that omittingbasically familiar material is rarely the answer tothe articulation problem. Rather the committeesees as more productive the following alternatives:

1) identifying successively higher levels ofpresentation;

2) changing emphases to take full advantageof students' preparation; and

3) le-allocating class time to permit inclusionof applications such as those in SMSGMathematics Through Science or inSMSG Mathematics And Living Things.

3

To assist in accomplishing these alternatives,this report lists references to topics from SMSGtexts from grades four through eleven with eachsection of the SMSG texts for junior high school,algebra, and geometry, and distinguishes strictlynew content from background material beingunified. The teacher is strongly urged to reviewthe sections cited, to draw examples from themfor reteaching and/or supplementation, to verifythat what may appear to be duplication of topicsmight be actually an ever widening spiral devel-opment of mathematical ideas.

The SMSG texts referred to are:E-4: Elementary School Mathematics,

Grade 4E-5: Elementary School Mathematics,

Grade 5E.6: Elementary School Mathematics,

Grade 6J-1: Mathematics for Junior High School

Volume 1, Parts I and IIJ-2: Mathematics for Junior High School

Volume 2, Parts I and II9: First Course in Algebra, Parts I and II

10: Geometry, Parts I and II11: Intermediate Mathematics,

Parts I andReferences to SMSG texts are in the form

(grade: chapter). (E-4:2), for instance, refers to thefourth grade text, Chapter 2.

COMMENTS ON ARTICULATINGSMSG MATHEMATICS

FOR JUNIOR HIGH. SCHOOLWITH STUDENTS' PRIOR SMSG

EXPERIENCES

For many years the junior high school yearswere the most barren ones in the mathematicscurriculum. Students could expect no more than.a rehash of arithmetic skills that they had learnedin elementary grades. With the advent of theSMSG Mathematics for Junior High School, arefreshingly new approach was given that gavenew impetus to the mathematics programs of theseyears. Subsequently, SMSG teams, well aware ofthe junior high context, wrote texts for fourth,fifth and sixth grades. This bulletin is written tohelp articulate the exploratory, informal work ofthe elementary years with the more unified andabstract content spiraling on through the juniorand senior high school mathematics curriculum.

There is more than enough material in the twovolumes of the SMSG Mathematics for JuniorHigh School to provide for a rich two year expe-rience. Although the learning process involves acertain amount of forgetting, and "refresher" ac-tivities must be provided even for students withprevious SMSG backgrounds, care must be takento avoid excessive repetition of basic concepts andskills. It is not necessary for pupils to repeat les-sons they have mastered before. Time is neededfor new ideas and new learning and every minutewe can continue to save is already spoken for. Onthe other hand, it would be unwise to omit mate-rial entirely since backgrounds of some studentswill be uneven. Thus, some overlap is highly desir-able. In fact, what may appear to be duplicationof topics is actually an ever-widening, spiral de-velopment of concepts and ideas, broad general-izations, developing mathematical ideas in a moreformal, mature manner than had been presentedinformally in earlier years.

In general, knowing that students come to theniwith previous SMSG preparation should causeteachers to provide a higher level of presentation,a changed emphasis, and a shortening of the timeallotted to the various chapters of the text. Thisbulletin is written to aid teachers of junior highschool mathematics in making judgments aboutthe amount of overlap and pacing in developingthe topics. Included in this section are referencesto topics which have been presented in previous

5

SMSG texts from grade 4 through grade 6, as wellas to applications and extensions of junior highcontent to be found in high school algebra andgeometry courses.

Suggested Time Schedule, Volume 1

Part IChapter

ApproximateNumber of Days

Recommendedin Commentary

Time Recommendedfor Students WithSeveral Years of

SMSG BackgroundTotal Number

of Days

1 7 5 52 1 5 203 104 10 304 1 5 10 405 12 12-15 556 1 7 20 757 13 10 858 13 10 95

PART II

9 18 2 11510 1210 12511 13 15 14012 11 10 15013 10 10 16014 5 5 1G5

Suggested Time Schedule, Volume 2Although students with several years SMSG

background have been introduced to many of thetopics of Volume 2, the development in everychapter here is so much more sophisticated thatthe times recommended in the commentary shouldnot be shortened. Needless to say, comments onarticulation are designed to complement, not sup-plant, Teacher's Commentaries.

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SMSG Mathematics for Junior High School,Volume 1, Part I

Chapter 1: (J-1:1)WHAT IS MATHEMATICS? (pp. 1-21)

Strictly motivational.

Chapter 2: (J-1:2)NUMERATION (pp. 21-61)2-1 Cavemen's Numerals (pp. 21-26)

Motivational.2-2 The Decimal System (pp. 26-30)

Rapid review of learnings from (E-4:2)*,(E-5:1) and (E-6:2).

2-3 Expanded Numerals and Exponential Nota-tion (pp. 30-33)

Prior to directing class review of this topic, theteacher is well advised to become familiar withthe extensive development of exponents in (E,6:1,2, 10) in order to appreciate how compressed thissection is. Work with exponents will continue in(J-2:3), where students study scientific notation.Supplementation now might well be drawn fromsectaotis on scientific notation in SMSG Mathe-matics Through Science, Part I, Chapter 2, andPart II, Appendix B.2-4 Numerals in Base Seven (pp. 3349),2 -5 Computation in Base Seven (pp. 39-48),2-6 Changing From Base Ten to Base Seven (pp.

48-52),2-7 Numerals in Other Bases (pp. 52-55), and2-3 The Binary and Duodecimal Systems (pp.

56-61)Numerals to bases other than ten were investi-

gated in (E-4:2) and (E-5:1) without benefit ofexponents. No work in bases other than ten ap-pears .at the sixth grade level. Consequently, thesections here combine at least two familiar topics(exponents and bases) in new ways.

Chapter 3: (j-1:3)WHOLE NUMBERS (pp. 67-105)3-1 Counting Numbers (pp. 67-70)

This section guides pupils to a new level ofmathematical maturity because it abstracts thenotion of one-to-one correspondence for the firsttime. The teacher might anticipate applying the

References to SMSG texts are in the form (Grade: chap-ter). (E-4:2), for instance, refers to the fourth grade text,Chapter 2.

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idea in the geometric setting of (J-1:4).3-2 Commutative Properties for Whole Numbers

(pp. 70474),3-3 Associative Properties for Whole Numbers

(pp. 75-78), and3-4 The Distributive Property (pp. 79-83)

In order to identify new abstractions to betaught in these sections, the teacher may retracefor himself explorations of properties in (E-4:3, 4,7), (E-5:2, 3), and (E-6:2). It is precisely becauseof those prior explorations that the seventh graderis ready for symbolic statements of properties. Stu-dents may profitably augment the problem setswith exercises of their own creation.3-5 Sets and the Closure Property (pp. 84-87)

The variety of sets familiar to SMSG studentsfrom (E-4:1, 5), (E-5:10), and (E-6:9) and earlyhints of closure in (E-4:3, p. 72; 7, p. 344) and(E-6:6, p. 371), should be known to the teacherbefore leading the class to the explicit idea ofclosure here for the first time.3-6 Inverse Operations (pp. 88-92)

SMSG cultivates a non-verbal awareness of in-verse operations in (E-4:3, 4), (E-5:3, 6), and (E-6:2). Before introducing it explicitly here, theteacher is urged to read those early sections as wellas look ahead to (J-2:2), where inverse operationsare first applied to equation solving.3-7 Betweenness and The Number Line (p. 93)

The number line is introduced.with countingnumbers in (E-4:3, 10), with non-negative nation -als in (E-5:6), with integers in (E-6:2, 4), and willbe used for the reals in (J-2:6). In the latter sectionbetweenness is basic to discussing irrational num-bers.3-8 The Number One (pp. 94-96) and3-9 The Number Zero (pp. 97-101)

The teacher should be familiar with earlier in-tuitive development in (E-4:3, 4), (E-5:2), and(E-6:2) of the properties of one and zero in orderto recognize the higher level of presentation here.Looking ahead, he can anticipate generalizationin (J-2:6), where zero and one are seen as identityelements. Appropriate supplementation with stu-dent-devised exercises is indicated.

Chapter 4: (J-1:4)NON-METRIC GEOMETRY (pp. 105-151)

Pupils who worked through SMSG Elementary

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School Mathematics in grades four, five, and sixhad extensive experience with intuitive geometry.Essential vocabulary and notation will be familiarto them. Properties formally stated in this chapterhave been explored on an informal basis. Sets andthe symbols pupils use in writing about them weredeveloped in the fourth grade and used in thefifth and sixth grades. Pupils have worked withsimple closed curves and with plane regions. Theyshould find much of this chapter a review on ahigher level of matet;a1 with which they haveacquaintance but probably only spotty recall.4-1 Points, Lines, and Space (pp. 106-108)

Pupils have used these words and ideas sincethe fourth grade. It should be sufficient to readand discuss the material for recall of ideas funda-mental to Property 1. This is an opportunity forthe teacher to listen while the pupils give evidenceof what they know.

4-2 Planes (pp. 108-113)Properties 2 and 3 formalize ideas suggested by

experimentation with models in grade four. Theteacher can rely on such remote experiences forlittle more than preparing pupils to accept thepresent level of formalism.

4-3 Names and Symbols (pp. 113-122)This section builds on material explored in

grade four. It may be read and discussed briefly,possibly with pupils demonstrating drawings andsymbols at the board.

4-4 Intersection of Sets (pp. 122-125)Symbols for ,nion and intersection were used

in (E-4:1,5) and (E-6:9).

4-5 Intersections of Lines and Planes(pp. 125-130)

Much of this material was explored in (E-4:5);it is brought to a more abstract level here.4.6 Segments (pp. 131-134)

Pupils have used the notation AB for segmentAB since (E-4:9), so they should move quicklythrough these pages.4-7 Separations (pp. 134-137)

Division of a plane by a line in the plane hasbeen talked about since grade four but the wordseparation has not been specifically mentioned.This subtle notion is being cultivated patientlyin anticipation of its significant application ingeometry (10-3).

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4-8 Angles and Triangles (pp. 138-141)The ideas of this section were developed intui-

tively in (E-4:5, 8), where pupils met points inte-rior to a region. The intersection of half planes,used to define the interior of an angle here and in(10-4), grows out of these earlier learnings.

4-9 One-to-One Correspondence (pp. 143-146)This section is an expansion of (J-1:3, 4).

4-10 Simple Closed Curves (pp. 147-150)Simple closed curves as such have not been dis-

cussed since their introduction in (E-4:5).

Chapter 5: (J-1:5)FACTORING AND PRIMES (pp. 151-189)5-1 Primes (pp. 151-155) and5-2 Factors (pp. 155-160)

The teacher is urged to familiarize himself withinvestigations in (E -5:2) and (E-6:2, p. 53) to iden-tify the new learnings emerging here as studentsuse exponents in prime factorization for the firsttime.

5.3 Divisibility (pp. 160-164)Aside from Testing 2, 3, and 5 as Factors of a

Number, (E-5:2), the student has not worked withtests for divisibility.

5.4 Greatest Common Factor (pp. ;64-169)This section rapidly reviews ideas from (E-5:2,

6) and (E-6:2), before unifying them with recentwork on exponents. Subsequent applications ofthe greatest common factor in simplifying non-negative rational numbers (J-1:6), signed rationals(J-2:1), and rational expressions (9:9, 12) morethan justify the spiraling of concepts seen here.

5-5 Remainders in Division (pp. 169-174)Skills and understandings not refreshed in

grade six after being developed in (E-4:7) and(E-5:2) are reviewed and advanced in this section.

5-6 Review (pp. 174-178)Time saved earlier in the chapter may profit-

ably be spent here.

5-7 Least Common Multiple (pp. 178-183)The least common multiple was investigated at

some length in (E-5:6) and very briefly mentionedin (E-6:2, 6). Here the skills and understandingsmust be reviewed thoroughly. The least commonmultiple of denominators will be required for

10

non-negative rational numbers in (J -1:6), forsigned rationals in 0-2:1), and for rational expres-sions in (9:9, 12).

Chapter 6: 0-1:6)THE RATIONAL NUMBER SYSTEM(pp. 189-243)

Thorough re-teaching of all the sections in thischapter is recommended in view of the variety andcomplexity of the learnings centered about therational number system. These topics are basic tostudy of percent (J-1:9), of signed rationals 0-2:1),and of rational expressions (9:9, 12).

Before undertaking such re-teaching via the ad-vanced approach here, the teacher may becomeinformed of the students' prior experiences byscanning the following content unified and ex-tended in this chapter.

TOPIC TEXTEquivalent Fractions (E-4:10), (E-5: 6), (E- 6 :2,6)Properties of Rational

Numbers (E-5:6), (E-6:2, 6)Reciprocals (E-6:2, 6)Using the Number Line

With RationalNumbers (E-5:6), (E-6:2)

Multiplying RationalNumbers (E-6:2)

Dividing RationalNumbers (E-6:6)

Adding and SubtractingRational Numbers (E-5:6, 10), (E-6:2)

Ratios (E-5:9)Decimal Notation (E-5:1, 6), (E-6:2, 6)Order (E-4:3, 10), (E-5:6)

Chapter 7 : U-1:7)MEASUREMENT (pp. 243-298)7 -I Counting and Measuring (pp. 243-249)

The idea of continuous quantities and the worddiscrete will be new. Properties of continuousquantities will also be new, but comparing areasand choosing units draw on preliminary work in(E -5:8).

7-2 Subdivision and Measurement (pp. 249-254)and

7-3 Subdividing Units of Measurement(pp. 254-260)

Subdivision is new but the vocabulary of meas-

11

urement as on p. 251 appeared in both grades fourand five.

7-4 Standard Units of Length (pp. 260-275)Standard units of measure were explored in

grades four and five, and now the division of inchand centimeter rulers are refined to 1/16 inchand 1 mm.

7-5 Precision of Measurement and the GreatestPossible Error (pp. 275-286)

These are strictly new ideas evolving from thepreceding four sections. Greatest possible error,cast within the context of science, appears inSMSG Mathematics and Living Things, Part 1,Chapter 2.

7-6 Measurement of Angles (pp. 287-298)Measurement of angles was carefully taught in

Chapter 7 of the fifth grade text. In grade six,pupils constructed perpendicular lines usingstraightedge and compass. Here such earlier les-sons are amplified.

Chapter 8: (J-1:8)AREA, VOLUME, WEIGHT AND TIME(pp. 299-345)

8.1 Rectangle (pp. 299-316)Some groundwork was laid with perimeters in

(E-4:9) and with areas in (E-5:8).8-2 Rectangular Prism (pp. 316-333)

Introductory work with solids, their faces, ver-tices, edges, and volumes in (E-4:8), (E-5:8), and(E-6:7) is augmented here in preparation for(J-1:10) and (J-2:11). Surface area and dimension-ality are new.

8-3 Other Measures (pp. 333-344)Computations with new units of measure.

SMSG Mathematics for junior High School,Volume 1, Part II

Chapter 9: (J-1:9)RATIOS, PERCENTS, AND DECIMALS(pp. 347-397)

This entire chapter is developmental, growingdirectly out of (J-1:6). Students with exclusivelySMSG backgrounds meet percent for the first time

12

here. Thorough teaching is demanded. Studentsshould be encouraged to devise supplementaryexamples of operations with decimals, uses ofequal ratios, and applications of percent.

Chapter 10: (J-1:10)PARALLELS, PARALLELOGRAMS,TRIANGLES, AND RIGHT PRISMS(pp. 397-463)10-1 Two Lines in a Plane (pp. 397-404),10-2 Three Lines in a Plane (pp. 404-409), and10-3 Parallel Lines and Corresponding Angles

(pp. 409-412)The new ideas in these sections build on the

work of (J-1:4, 8).

10-4 Converse (Turning a Statement Around)(pp. 412-416)

The teacher is referred to (E-4:6), where "if-then" thinking was used on a modest scale. The"if-then" form which was not employed explicitlyagain until this little section enhances the stu-dents' preparation for algebraic and geometricproofs in (9:7, 8) and (10:3).

10.5 Triangles (pp. 416-424) and10-6 Angles of a Triangle (pp. 424-431)

These sections, just as the first three of the chap-ter, draw on (J-1:4, 8).

10-7 Parallelograms (pp. 431-438)This is new content, relying upon section 3

above.

10-8 Areas of Parallelograms and Triangles.(pp. 438-448)

The teacher is advised to scan in (E-5:8) the ex-ploratory activities which reprPsent the students'prior experience with areas of parallelograms andtriangles. The teacher needs to be alert to thesophistication of the presentation of this section,for this material is extended in (J-2:11).

10-9 Right Prisms (pp. 448-462)This section expands upon (J-1:8) and is pre-

paratory to (J-2: 1 1).

Chapter 11: (J-1:11)CIRCLES (pp. 463-525)11-1 Circles and the Compass (pp. 463-469)

Pupils are familiar with use of the compass and

13

with such terms as center and radius from (E-4:5,9) and (E-5:9). One lesson is sufficient for a rapidreview.

11-2 Interiors and Intersections (pp. 469-475)Intersections of circles with other circles, lines,

and triangles in (E-6:9) and (I-1:4, 5) preparedstudents for definitions of interior and exterior ofcircles here. The term concentric is new.

11-3 Diameters and Tangents (pp. 475-481)This new material is pointed toward (10:13).

11-4 Arcs (pp. 481-485)This section synthesizes familiar ideas of sep-

aration (J-1:4) and arcs (E-5:4) and (E-6:9) to pro-duce better understanding.

11-5 Central Angles (pp. 486-492)In (E-6:9) pupils worked briefly with central

angles.

11-6 Circumference of Circles (pp. 492-500)This is new content, induced from Exercise 7 of

the last section.

11-7 Area of a Circle (pp. 500-508)The empirical approach to area through count-

ing squares, familiar from (E-5:8) and (E-6:7), cul-minates in the formula for the area of a circlehere. In (J-2:11) and (10:15), the area of a circle isseen as the limit of the areas of inscribed polygons.

11-8 Cylindrical Solids Volume (pp. 509-514)The teacher is urged to become familiar with

the exploration of volumes in (E-6:7) to appre-ciate just how concise the present section is.

11-9 Cylindrical Solids Surface Area(pp. 514-519)

Extends the thinking developed in (J-1:8).

11.10 Optional: Review of Chapters 10 and II(pp. 519-525)

If the teacher has succeeded in utilizing the stu-dents' SMSG background sufficient/y, ample timewill be available for these enrichment-type activi-ties as well as some of the uses of circles andcylindrical solids found in SMSG Mathematicsand Living Things, Part 2.

Chapter 12: (J -1:12)MATHEMATICAL SYSTEMS (pp. 527-579)

Modular arithmetic here affords extension and

14

reinforcement for some of the more abstract no-tions developed in earlier chapters of this text asbackground for algebra (9:3, 8).

Chapter 13: (J -1: 13)STATISTICS AND GRAPHS (pp. 579-609)

To appreciate fully the maturity of the mate -rials in this chapter, the teacher is advised to in-spect (E-6:8), the only prior SMSG content directlyrelated to statistics. Here basic ideas and skills arerefreshed and amplified. More extensive utiliza-tions of graphing and statistical analysis of experi-mental data are available in SMSG Mathematicsand Living Things.

Chapter 14: C/-1:14)MATHEMATICS AT WORK IN SCIENCE(pp. 609-623)

Most of the students' mathematics up to thispoint have found applications internally doingmore mathematics. This chapter offers an externalapplication which may very well lead to consider-ing parts of Polya's Mathematical Methods inSctencel, and the SMSG volumes MathematicsThrough Science, in anticipation of increasinglyformal considerations of inverse variation in(J -2:9) and (9:14).

SMSG Mathematics foi Junior High School,Volume 2, Part I

Chapter 1: (J -2:1)RATIONAL NUMBERS ANDCOORDINATES (pp. 1-50)1-1 The Number Line (pp. 1-7)

Uses of the number line to show addition andsubtraction of whole numbers (E-4:3), of positiverationals (E-5:6), of integers (E-6:4), and to eluci-date betweenness (I-1:3) are consolidated here inpreparation for subsequent work in 0-2:6), (9:1),and (10:2).

1.2 Negative Rational Numbers (pp. 7-12)SMSG students meet negative rationals here for

the first time.

' Poiya, George. Studies in Mathematics, Volume XI, Mathe-matical Methods in Science, Stanford: School MathematicsStudy Group. 190.

15

1-3 Addition of Rational Numbers (pp. 12-21)Opposites appeared in (E-6:4) with integers

only; the additive inverse of a rational number isnew at this point. Addition of signed rationalnumbers is new.

1-4 Coordinates (pp. 21-30)An introduction to graphing on a line and in

a plane was presented in (E-6:5), consequentlyonly a. brief review is needed here. (Exercises 1-4cserve as an adequate review.)

1-5 Graphs (pp. 30-34)Graphing inequalities in a plane is new. The

subject is expanded in both the ninth and tenthgrade texts, consequently only an introduction isoffered here.

1-6 Multiplication of Rational Numbers(pp. 34-41)

This is new material. The use of the distribu-tive property as described on page 38 mesheswith experiences in the elementary texts, e.g.,(E-6:2), and in (J-1:6).

1-7 Division of Rational Numbers (pp. 42-46)Reading the limited approach to this topic in

(J-1:6) will aid the teacher in identifying newaspects of the topic presented here.

1-8 Subtraction of Rational Numbers (pp. 46-50)Subtraction of integers was taught in (E-6:4);

however, it probably needs to be re-taught in fulldetail here before moving on to subtraction ofsigned rational numbers.

Chapter 2: (J -2:2)EQUATIONS (pp. 51-110)2-1 Writing Number Phrases (pp. 51-57)

The concept of a number sentence is an integralpart of the elementary SMSG texts, recurring in(E-4:3), (E-5:3), and (E-6:4). This section shouldrequire only brief trmtment, possibly little morethan a discussion of some of the exercises.

2-2 Writing Number Sentences (pp. 58-75)This section provides a background for (9:3, 4)

and should be taught as an introduction, not forcomplete mastery. Symbols expressing inequali-ties, employed only incidentally in the elementarytexts, are essentially new.

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2-3 Finding Solution Sets (pp. 75-90).This is the student's first formal approach to

finding the solution set of an open sentence. Theteacher will benefit by reading (9:6) and (9:7) tosee how the subject is taught the next year. Don'texpect mastery here.

2-4 Solving Inequalities (pp. 91-93)Use of the addition property of inequality, new

to students here, will be studied again in (9:8)with the multiplication property of inequality.Treat this section as introductory.

2-5 Number Sentences With Tivo Unknowns(pp. 93-107)

The concepts here are new and difficult and willbe studied in much greater detail in algebra. Evenwell-prepared students will be challenged. This isagain introductory only.

Chapter 3: (J -2:3)SCIENTIFIC NOTATION, DECIMALS, ANDTHE METRIC SYSTEM (pp. 111-155)3-1 Large Numbers and Scientific Notation

(pp. 111-117),3-2 Calculating With Large Numbers

(pp. 117-120),3-3 Calculating With Small Numbers

(pp. 120-125),3-4 Multiplication: Large and Small Numbers

(pp. 125-128), andDivision: Large and Small Numbers(pp. 128-133)

These sections provide the student's first en-counters in regular SMSG texts with scientific no-tation. They jog his memory concerning positiveexponents from (E-5:1), (E-6:1), and U-1:2,5), in-troduce negative exponents, and reinforce com-putational skills requisite for (9:10).

3-6 Use of Exponents in Multiplying and Divid-ing Decimals (pp. 133-137)

The _treatments of decimals in (E-6:2) andU-1:9) gave background for these new skills.3-7 The Metric System; Metric Units of Length

(pp. 137-145),3-8 Metric Units of Area (pp. 145-148),3-9 Metric Units of Volume (pp. 148-150), and.3-10 Metric Units of Mass and Capacity

(pp. 150-155)

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These last four sections pursue topics whichspiral through (E-4:9), (E-5:8), (E-6:7), and U-1:7).

Chapter 4: (J -2:4)CONSTRUCTIONS, CONGRUENTTRIANGLES, AND THE PYTHAGOREANPROPERTY (pp. 157-214)4-1 Introduction to Mathematical Drawings and

Constructions (pp. 157-162)This section reviews and maintains the skill of

using a protractor studied in (E-5:7) and (J.1:7).

4-2 Basic Constructions (pp. 162-171)Straightedge and compass constructions were

treated in (E-5:4) and (E-6:9). Review the presentsection very rapidly. (Exercises 4-2b alone may beadequate.) Constructions are again studied in(10:14).

4-3 Symmetry (pp. 172-176)Line symmetry was discussed in (E-6:5). Only a

brief review is needed, allowing time to elaborateon symmetry with respect to a point in ProbleM 7of Exercises 4-3.

4-4 Congruent Triangles (pp. 176-183)This section advances the student's thinking

toward the goal of formal geometric proof at thetenth grade level. The teacher is urged to becomefamiliar with investigation of conditions for con-gruence of triangles in (E-5:4) and of side andangle relationships of triangles in (E-6:3). Againstthis ample background, it should be possible tocover the material rapidly.

4-5 Showing Two Triangles to be Congruent(pp. 184-192)

Teach this section intuitively; few students aremature enough yet to benefit from the formaltreatment given in the tenth grade. Constructinga perpendicular from a point to a line is refinedhere over the simple techniques of (E-6:9).

4-6 The Right Triangle (pp. 192-201)Although right triangles were met in (E-4:8)

and (E-6:3), this material is new. Develop it indetail. Because the treatment of square roots in(9:11) is succinct, give supplementary work withsquare root tables now.

4-7 One Proof of the Pythagorean Property(pp. 202-205)

This topic should be presented to every group

18

with a background in SMSG elementary mathe-matics. A non-algebraic proof adaptable to over-head projectors appears in SMSG Introduction toSecondary School Mathematics, Volume 2, Part. II,Chapter 19.

4-8 Quadrilaterals (pp. 206-208)Consolidates and furthers studies from (E-4:5),

(E-6:9), and 0-1: 1 oy.

4-9 Solids (pp. 208-214)Sketching planes and three-dimensional figures

is new. It will find its immediate application in(J -2:11) and thereafter in tenth grade geometry.

Chapter 5: (J -2:5)

RELATIVE ERROR (pp. 215-234)New aspects of greatest possible error (J -1:7) and

of scientific notation 0-2:3) are probed.

Chapter 6: (J-2:6)

REAL NUMBERS (pp. 235-279)6-1 Review of Rational Numbers (pp. 235-240)

Material from (J -2:1) are reviewed here prep-aratory to investigating new properties of therationals.

6-2 Density of Rational Numbers (pp. 240-245)The teacher may want to read the treatment of

this new concept in (9:1).

6-3 Decimal Representations for the RationalNumbers (pp. 245-250)

In (E-2:6) and (j-1:9) this topic was introducedbriefly. The development here is in greater detailand contains many new ideas.

6-4 The Rational Number Corresponding to aPeriodic Decimal (pp. 250-254)

This is new material. The students' strong back-ground in factoring from (E-5:2), (E-6:2), and(J -1:5) should make explanation of terminatingdecimals easily understood.

6-5 Rational Points on the Number Line(pp. 255-257)

This new material is fundamental to the studyof rational approximations of irrational numberslater in this chapter and in (9:11).

19 -

6-6 Irrational Numbers (pp. 257-262) and6-7 A Decimal Representation for \a-

(pp. 262-267)New and difficult content. The term irrational

is new; the number 7 was approxima ted in (J-1:11,pp. 496-491 via an argument similar to the onehere for 1/2 without benefit of the vocabulary ofirrational numbers. Guard against pressing formastery of the proof; it recurs in (9:1I). A some-what simpler form appears in SMSG Introductionto Secondary School Mathematics, Volume 2, PartII, Chapter 20.

6-8 Irrational Numbers and the Real NumberSystem (pp. 267-275) and

6-9 Geometric Properties of the Real NumberLine (pp. 276-279)

This new and difficult content should be taughtas introductory, not for mastery; it is simply back-ground for intensive investigations appropriatelyreserved to (9:11) aril (10:11).

SMSG Mathematics for Junior High School,Volume 2, Part II

Chapter 7: (J-2:7)PERMUTATIONS AND SELECTIONS(pp. 281-310) and

Chapter 8: ( J-2: 8)PROBABILITY (pp. 311-345)

The exploratory nature of activities in thesechapters should not be obscured by occasionalhints at formalization. Such formalization is rele-gated to the eleventh grade (11:14). Varied sup-plementary activities are provided in SMSG Prob-ability for Intermediate Grades and SMSG Intro-duction to Probability, Parts I and 2.

Chapter 9: (J-2:9)SIMILAR TRIANGLES AND VARIATION(pp. 347-416)9-1 Indirect Measurement and Ratios

(pp. 347-358) and9-2 Trigonometric Ratios (pp. 358-365)

(E-5:9) and (J-1:6) constitute the student? priorexperience with ratio; (E-6:6) and (J-2:1,2), their

20

experience with coordinates in a plane. The unifi-cation and extensions here are pointed towardcultivating a feeling for trigonometric ratios, asopposed to formal manipulation which is appro-priately postponed until (10: Appendix X) and(11:10).

9-3 Reading a Table (pp. 365-374)Skills in reading tables are being expanded

here. Interpolation is taught directly in (11:10)for the first time.

9-4 Slope of a Line (pp. 374-379)This section grows out of the one immediately

preceding it and provides some foundation forfurther study in (9:14). Slope is used as a tool inanalyzing experimental data in SMSG Mathe-matics Through Science, Part I, Chapter 4.

9-5 Similar Triangles (pp. 379-384) and9-6 Scale Drawings and Maps (pp. 384-391)

The new concept of similarity is founded incongruence, an idea familiar from (E-6:3), (J-1:7),and (J-2:4). These sections advance the students'facility with ratio by associating proportional andscale with ratios.

9-7 Kinds .a Variation (pp. 392-395),9-8 Direct Variation (pp. 396-400),9-9 Inverse Variation (pp. 401-407), and9-10 Other Types of Variation (Optional)

(pp. 407-416)Mathematical ideas continue to spiral here,

uniting notions of slope from section 4 abovl,elaborating on The Scientific Seesaw from (J-1:14), and extending understanding in the very kindof applications teachers of high school science ex-pect their students to have in mathematics classes.SMSG Mathematics Through Science, Part II;Chapters I, 2, and 3 contain abundant supple-mentary material.

Chapter 10: ( J,2: 10)NON-METRIC GEOMETRY (pp. 417-449)

The teacher may profitably skim through non-metric geometry in (J-1:4) and explorations withsolids in (J-1:8, 10, 11) before moving into thisdevelopmental chapter. Looking ahead, SMSGGeometry integrates three-dimensional work withplane geometry on the supposition that the foun-dation from these investigations at the eighth

21

grade level will survive a year's disuse while thestudent attends to algebra. Student-constructedmodels are indispensable to the present develop-ment.

Chapter 11: (J-2:11)VOLUMES AND SURFACE AREAS(pp. 451-510)

Previous study of this topic occurred in (J-1:8,10). Computational practice is more than an inci-dental bonus for working through this chapterever on guard against pre-empting the formalismemphasized in tenth grade geometry (10:16).

Chapter 12: (J-2:12)THE SPHERE (pp. 511-544)

The progression from the familiar to the new israpid here: the exploratory emphasis must bepreserved, with abundant use of concrete models.The benefits of the present investigations will beevident when spheres are formally treated in(10:16).

Chapter 13: (J-2:13)WHAT NOBODY KNOWS ABOUTMATHEMATICS (pp. 545-579)

Time recommendations for the eighth gradepresuppose reliance on the prior SMSG expe-riences cited throughout this report. It is conso-nant with the spirit of SMSG to resist anytemptation to prolong practice beyond the recom-mendations and thereby to assure inclusion ofthis provocative final chapter.

22

COMMENTS ON ARTICULATING FORSMSG FIRST COURSE IN ALGEBRA

WITH STUDENTS' PRIOR SMSGEXPERIENCES

The greatest responsibility of a classroomteacher is probably that of making judgments.The teacher must make judgments about stu-dents, grades, selection of subject matter, and in-structional procedures. Certainly, one of the mostimportant phases in the entire evaluation processis the articulation of content as well as the levelof presentation.

Articulation which involves the transition fromone mathematics course to the next, has alwayspresented a vexing problem to mathematics teach-ers. It is essential that teachers use diagnostic pro-cedures in order to avoid duplication of contentwhich students have met in previous SMSG texts'.Furthermore, because the learning process in-volves a certain amount of forgetting, "refresher"activities must be provided even for students withprevious SMSG backgrounds. It is not .surprisingthat students become bored when supposedly newcourses and topics in mathematics turn out to bedull rehash of topics which they have "discovered"and studied years earlier. On the other hand, itwould be unwise to omit material entirely sincebackgrounds of some students will be uneven.Thus, some overlap is highly desirable. In fact,what may appear to be duplication of topics isactually an ever-widening spiral development ofconcepts and ideas. The Teacher's Commentary ofthe SMSG First Course in Algebra continuallyemphasizes the spiral approach.

At present, teachers are finding that studentsbeginning the study of First Course in Algebra arebetter prepared than they were formerly becauseof previous SMSG background. In pacing and de-veloping the course, teachers must take into ac-count that some of the material in SMSG FirstCourse in Algebra is not new.

Specifically, SMSG introduces sets, the numberline, and some of the properties of x; 4-, asearly as the fourth grade; factors and primes ap-pear at the fifth grade; exponents and negativenumbers at the sixth. The real number system isinvestigated at some depth at the eighth gradelevel.

In general, knowing that students come to themwith previous SMSG preparation should causeteachers to provide a higher level of presentation,

23

a changed emphasis, and a shortening of the timeallotted to some of the early chapters of the text.

This section is written to aid teachers of FirstCourse in Algebra in making judgments about theamount of overlap and pacing in developing thetopics. Included in this bulletin arc detailed ref-erences to topics which have been presented inSMSG texts for grades 4 through 8, and which willbe presented in SNISG algebra and geometry texts.

Suggested Time Schedule

Part IChapter

ApproximateNumber of DaysRecommendedin Commentary

Time Recommendedfor Students WithSeveral Years of

StilSG BackgroundTotal Number

of Days

1 6 1-2 22 8 4 63 18 10 164 10 3 195 10 5 246 10 2-3 277 6-10 378 6 439 12 55

PART It

10 10 6511 12-15 8012 20-25 10513 15-20 125 .14 5 13015 5-10 14016 10-15 15517 10 165

Rapidly covering the first five chapters, whichdraw heavily on pupils' prior SMSG experience,affords more than adequate time for new topicssuch as rational expressions, radicals, and com-pleting the square.

24

SMSG First Course in Algebra,Part I

Chapter 1: (9:1)SETS AND THE NUMBER LINE (pp. 1-18)1-1 Sets and Subsets (pp. 1-6)

Such terms as set, elements, union, intersection,and subsets, as well as the symbols { ), it,, U, andfl, appear as early as (E-4:1) and (E-5:2, 6, 10).Set language is also used in (E-6:4, 5, 9), (J-1:3, 6),and (J-2:1, 4, 6) in developing the integers, ra-tionals, and real numbers. Non-metric geometry(J-1:4) uses sets, while finite and infinite sets areidentified in (J-1:12). The present section reviewsonly a few basic notions which, under the teacher'sguidance, may prompt students to recall moredetails.

1-2 The Number Line (pp. 7-14)The number line is used throughout SMSG

texts: it is introduced in (E-4:3), and is used tointerpret integers and rational numbers and theiroperations in (E-5:6), (E-6:4, 5), (1-1:3), and (J-2:1),and to work with real numbers in (J-2:6). Thissection, restricted as it is to the non-negative realnumbers, may appear trivial unless judiciouslysupplemented, by student- constructed examples,for instance.

1-3 Addition and Multiplication on the NumberLine (pp. 14-18)

The number line was used to describe opera-tions on integers in (E-6:4) and rationals in (E-5:6),(E-6:2), and (J-2:1). Students with several yearsSMSG background require but a review here.

Chapter 2: (9:2)NUMERALS AND VARIABLES (pp. 19-23)2.1 Numerals and Numerical Phrases (pp. 19-23)

The distinction between number and numeralbegan in (E -4:2). U -1:2) dealt with these terms inan explicit fashion. Numerical phrases and sen-tences appeared in (E-4:3), (E-5:9), and (J-2:2).Teachers should be able to go through this sectionas a rapid refresher exercise.

References to SMSG texts are in the form (Grade: chap-ter). (E-4:1), for instance, refers to the fourth grade text,Chapter 1.

25

2-2 Some Properties of Addition andMultiplication (pp. 23-28)

Binary. .operations defined on a set first .appearin (J -1:12) through a discussion of abstract mathe-matical systems. Commutative and associativeproperties have been stated in a general fashionfor addition and multiplication of counting num-bers, whole numbers, integers, and rational num-bers in (E-4:3, 4), (E-5:2, 6, 10), (E-6:2, 4, 6), (J-1:3,6), and (J -2:1). Mathematical Systems (1-1:12) andReal Numbers (J-2:6) present these propertiesmore formally. The review here concentrates onnon-negative reals.

2-3 The Distributive Property (pp. 29-34)The distributive property of multiplication

over addition received considerable emphasis in(E-4:4), (E-5:3, 10), (E-6:2), (J -1:6), and (J-2:1)through numerical examples. It was handled moreformally in (J -2:6). Its re-teaching here is basic tothe concept of variable in the next section.

2-4 Variables (pp. 34-39)The idea of a variable has been used informally

since its introduction in (E-4:3), where, e.g., n rep-resents the number which makes the sentence "10

n = 30" true. This treatment is repeated in(E-4:4, 6, 7, 10), (E-5:3, 5, 9, 10), and (E-2:1, 2, 4,5, 6, 7, 10). The seventh and eighth grade SMSGtexts continue informal treatment of variables.The terms variable and domain will be new tostudents.

Chapter 3: (9:3)SENTENCES AND PROPERTIES OFOPERATIONS (pp. 41-76)3-1 Sentences, True and False (p. 41)

Mathematical sentences are introduced veryearly in SMSG texts and become an integral partof the texts. In (E-4:3) students are asked to judgesentences involving equations and inequalities astrue or false, just as they are here in brief reviewexercises.

3-2 Open Sentences (pp. 42-44)Open Sentences are treated informally in (E-4:3,

4, 6, 7; 10), (E-5:3, 6, 9, 10), (E-6:2, 4, 6, 10), 0-1:3,6), and (J-2:2). Although the present review is castwithin the context of the non-negative real num-bers, the teacher may be well advised to exceedthat limitation by admitting any negative real

26

numbers students recall as they attack ProblemSet 3-2c. for example.

3-3 Truth Sets of Open Sentences (pp. 45-47)The phrase truth set of an open sentence will

be new to students, although they have been find-ing truth sets informally since (E-4:3). On page 60of U-2:2) truth sets of open sentences involvingequations and inequalities are treated explicitlybut there they are called solution sets. Such priorexperiences are extended in this section andshould be recalled by specific examples.

3-4 Graphs of Truth Sets (pp. 47-48)SMSG students 'explored graphing extensively

in U-2:1, 2); the present one-page review of thebare minimum is requisite to continued reinforce-ment and extension of concepts through the nextsix sections.

3.5 Sentences Involving Inequalities (p. 49)Inequalities are developed gradually in (E-4:2,

10), (E-5:9), (E-6:4), U-1:3), and 0-2:2); here expo-sition and exercises reinforce the basic vocabularywhich may have been forgotten because of disuse.

3-6 Open Sentences Involving Inequalities(pp. 50-51)

Graphing inequalities is fascinating for manystudents; nonetheless, the skills required may bereadily forgotten. The teacher, familiar with thedevelopment of graphing inequalities with ra-tionals in (J-2:11, 2), can identify the few ideasbeing reviewed here and in section 3-8 below inanticipation of their extension to graphing in-equalities with the reals in (9:5).

3-7 Sentences With More Than One Clause(pp. 52-53)

Compound sentences formed by using the con-nectives and and or are introduced very briefly in0-2:2, pp. 68-70), along with a few graphs of thetruth sets of easy compound sentences.The teachercan rely on little from that previous experienceexcept that students may be vaguely aware of thetopic, ready for thorough re-teaching. Encouragestudents to extend the problem sets to inctudeexamples of compound sentences using any real.number.

3-8 Graphs of Truth Sets of Compound OpenSentences (pp. 54-55)

The subtle logic of these sentences warrants

27

handling this section simply to reinforce all priorrelated topics.

3-9 Summary of Open Sentences (pp. 56-57)The summary at this time provides a crystalli-

zation of i -eas of the first eight sections and aspringboard into the next sections on propertiesof addition, and multiplication. Students withSMSG background should be successful in devis-ing and graphing open sentences beyond thoseprovided in the text.

3-10 Identity Elements (pp. 57-60)Identity elements for addition and multiplica-

tion are introduced informally in (E-4:3, 5),(E-5:6), (E-6:2, 6), (J-1:3, 6), and (J-2:6) as theProperty of Zero and the Property of One. Iden-tity elements in an abstract mathematical systemare discussed in (J-1:12). The quantifier for everya introduces new rigor in this section. Least com-mon multiple is treated in (E-5:6) and (J-1:5, 6)but with simpler applications than developedhere.

3-11 Closure (pp. 60-61)Closure for whole numbers is presented infor-

mally in (E-4:3, 4). Discussion of this property forrational numbers appears in (E-5:6) and (E-6:2, 6).The concept of a closed set under an operationand the term closure appear in (J -1:3, 12). Closurefor the reals is mentioned in (J-2:6) but withoutextensive rationalization and applications offeredin sections 3-12 and 3-13 below; i.e., they are con-fined to non-negative real numbers.

3-12 Associative and Commutative Properties ofAddition and Multiplication (pp. 61-66)

This section consolidates concepts previouslyinvestigated and reiterated in a variety of settings(see 2-2 and 3-11 above).

3-13 The Distributive Property (pp. 66-71)The language of algebra is employed here for

the first time to formalize and generalize muchprevious study (see 2-3 above) less abstractly de-rived from arithmetic. These topics have been soeffectively spiraled that the student of SMSG willbe convinced of the validity of the distributiveproperty and ready to apply it.

3-14 Summary: Properties of Operations onNumbers of Arithmetic (pp. 71-76)

The .11.1M/narl of properties is for reference, notdrill. A guide for teaching emphasis is implicit in

28

the observation that less than one page is givento listing the properties; four pages are devoted toapplications and extensions.

Chapter 4: (9:4)OPEN SENTENCES AND ENGLISHSENTENCES (pp. 77-96)

Against the SMSG students' use of open phrasesand sentences consistently since grade four, thissection may cloy unless the teacher covers it at afast pace and encourages pupils to utilize theirbackgrounds in posing comparable problems.

Chapter 5: (9:51THE REAL NUMBERS (pp. 96-.120)5-1 The Real Number Line (pp. 96-101)

The negative integers, opposites, and the num-bering of the left half of the number line weretaught initially in (E-6:4), negative rational num-bers in (J-2:1), real numbers in (J-2:6). This sec-tion is essentially review and unification.

5-2 Order on the Real Number Line(pp. 102-107)

This is new content, building on many earlierpresentations, particularly those of 3-5 above.

5-3 Opposites (pp. 108-112)The idea of opposites was presented for integers

in (E-6:4) and for rational numbers in (J-2:2).Relatively little difficulty should be encounteredin a rapid review of the topic.

5-4 Absolute Value (pp. 113-120)The concept of absolute value is new.

5-5 Summary (p. 118)Numerical illustrations given by the student

may reinforce the principles summarized fromthis chapter.

Chapter 6: (9:6)PROPERTIES OF ADDITION (pp. 120-144)6-1 Addition of Real Numbers (pp. 121-124),6-2 Definition of Addition (pp. 124-129), and6-3 Properties of Addition (pp. 129-132)

Students of SMSG had graphing and computa-tional practice adding signed numbers in (E-6:4)and (J-2:1). Therefore, the initial sections of thischapter should be covered rapidly to restore skill

29

while infusing the new interests of handling ab-solute values and looking at more formal state-ments of addition properties.

6-4 The Addition Property of Equality(pp. 132-135) and

6-5 The Additive Inverse (pp. 135-140)The addition property of equality was intro-

duced and applied to the solution of equations in(J-2:2). The concept of the inverse of an elementunder a binary operation was introduced in thediscussion of abstract mathematical systems (J-1:12). Additive inverses for rational numbers wereidentified and applied to the solution of equa-tions in (J-2:1). As a consequence of these expo-sures, SMSG students may solve many of the exer-cises in these sections by inspection and supple-ment them with examples of their own devising.

Chapter 7: (9:7)PROPERTIES OF MULTIPLICATION(pp. 145-183)7-1 Multiplication of Real Numbers

(pp. 145-150),7-2 Properties of Multiplication (pp. 150-156),

and7-3 Use of the Multiplication of Properties

(pp. 156-159)These sections draw together such previous

topics for SMSG students as multiplication ofsigned numbers in (E-6:4) and (J-2:1), phrases andopen sentences (9:2), and properties of multipli-cation (9:3). Although many of the exercises inthese first three sections lend themselves to oralwork and solution by inspection during rapid re-view, the content of the balance of Chapter 7demands a slower pace.

7-4 Further Use of the Multiplication Properties(pp. 159-162)

Exponents on numerical bases are familiar toSMSG students from (E-6:1), (J-1:5), and (J-2:3).The teacher might use examples from these textsin introducing the new content of the presentsection.

7-5 Multiplicative Inverse (pp. 162-165),7-6 Multiplication Property of Equality

(pp. 165-166),7-7 Solutions of Equations (pp. 167-172), and7-8 Reciprocals (pp. 172.-180)

30

Multiplicative inverses and reciprocals occur in(E-6:2), U-1:6), and (J-2:1). The inverse of an ele-ment under a binary operation was discussed forabstract mathematical systems in 0-1:12). Themultiplicative property of equality was appliedto the solution of equations in (J-2:2). It is sug-gested that teachers rely on these preliminary ex-periences for little more than facilitating furtherpresentations in the present sections.

7-9 The Two Basic Operations and the Inverse ofa Number Under These Operations(pp. I 80-183)

This is a sophisticated summary of Chapters 6and 7: in addition, multiplication, and their in-verse operations.

Chapter 8: (9:8)PROPERTIES OF ORDER (pp. 185-208)

Immediately following a brief review in section8-1, the chapter emerges as strictly developmental,building new commit directly on material fromprevious chapters of the First Course. Here again,encouraging students to handle as much of thepractice material as they can orally not only capi-talizes on their previous understanding but alsoalerts them to methods of attacking written prob-lems demanding more structured approaches.

Chapter 9: (9:9)SUBTRACTION AND DIVISION FOR REALNUMBERS (pp. 209-245)9-1 Definition of Subtraction (pp. 209-212) and9-2 Properties of Subtraction (pp. 212-218)

The definition of subtraction for reals here isan extension of the definition of subtraction forrationals 0-2:1). Hence, the student with SMSGbackground will see these sections as aids to re-calling and strengthening earlier understandingsthrough rapid review leading into new, time-con-suming, exercises.

9.3 Subtractions in Terms of Distance(pp. 219-223)

This is new, drawing on the student's familiar-ity with the number line and on his recent experi-ences with absolute value (section 4 of chapter 5).

9-4 Division (pp. 223-229)Division of rational numbers in (E-6:2), (J-1:6),

31

and (J-2:1) precedes the extension in this sectionof division to the real numbers. Oral work on thesimple practice materials will serve to renew skillsprior to attempting the new and difficult exercises.

9-5 Common Names (pp. 229 -233) and9-6 Fractions (pp. 233-241)

The introductory remarks in these sections ap-peal to earlier understandings concerning rationalnumbers spiraling through (E-5:6), (E-6:2), (J-1:6),and (J-2:1). Students with SIVISG backgrounds canbe expected to proceed readily from the review-type numerical problems to the new algebraic ra-tional expressions, where time saved earlier canproductively be spent now.

SMSG First Course in Algebra,Part II

Chapter 10: (9:10)FACTORS AND EXPONENTS (pp. 247-282)10-1 Factors and Divisibility (pp. 247-251)

Factoring, with tests for factors, is introduced in(E-5:2). In this section, formal definitions of fac-tor and .proper factor are generalizations of defi'salons in, (J-1:5). Review these numerical exer-cises rapidly.

10-2 Prime Numbers (pp. 252-254)The terms prime and composite were encoun-

tered in (E-5:2) and (J-1:5); the Sieve of Eratos;thenes, in (J-1:5). This-numerical content is nottime - consuming. l.10-3 Prime Factorization (pp. 255-257)

The Fundamental Theorem of Arithmetic ap-peared as. the Unique Factorization Property ofCounting Numbers in (J-1:5), with the strictlynumerical applications reiiewed briefly here.

10-4 Adding and Subtracting Fractions(pp. 258-261)

The least common multiple and its application_to adding fractions appeared in (E-5:6) and (J-1:5).

' Addition and subtraction of rational numbersspiraled upward in (J-1:8) and (J-2:1). In the pres-ent section, students progress from those earlier,arithmetic manipulations to simple algebraicfractions.

32

10-5 Some Facts About Factors (pp. 261-266)Although divisibility notions from (J-1:5) are

extended and formalized here, the numerical em-phasis requires little time.

10-6 Introduction to Exponents (pp. 266-269)and

10-7 Further Properties of Exponents(pp. 269-282)

These sections develop new content as theymove rapidly from the numerical emphasis ofsection 7-4 above to algebraic expressions.

Chapter 11: (9:11)RADICALS (pp. 283-31.1)

New content. Pre-algebra SMSG texts provideless reiteration of the mathematics of radicals thanof many other topics; consequently, the teacher isadvised to become familiar with the modest intro-ductions to square root (J-2:4) and irrational num-bers (J-2:6)-before-beginning- this- developmentalchapter.

Chapter 12: (9:12)POLYNOMIAL AND RATIONALEXPRESSIONS (pp. 313-375)

The predominantly new content of this chapterreviews and significantly extends the mathematicsof the prekeding eleven chapters.

Chapter 13: (9:13)TRUTH SETS OF OPEN SENTENCES(pp. 377-404)

The chapter synthesizes much fundamentalmathematics from the students' background inSMSG. The idea of equivalent open sentences wasintroduced in (J-2:2), but there the term equiva-lent equations was employed. The addition prop-erty of equality appears in (J-2:2) and (9:6), themultiplication property of equality in (J-2:2) and(9:7), opposites of integers in (E-6:4) and (J-2:1),.additive tilverses for rational numbers in (J-2:1),multiplicative inverses in (J-2:6), reciprocals in(9:7). =All of this background, and more, exists;nonetheless, some re-teaching is implicit.

Chapter 14? (9:14)GRAPHS OF SENTENCES IN TWOVARIABLES (pp. 405-463)14-1. The Real Number Plane (pp. 405-411),

33

14-2 Graphs of Open Sentences With TwoVariables (pp. 411-422), and

14-3 Slopes and Intercepts (pp. 423-439)(E-6:5) and 0-2:1, 2, 9) provided successive in-

crements to skills and understandings being am-plified in these sections.

14-4 Graphs of Open Sentences InvolvingIntegers Only (pp. 440-447),

14-5 Graphs of Open Sentences InvolvingAbsolute Value (pp. 448-463),

Chapter 15: (9:15)SYSTEMS OF EQUATIONS ANDINEQUALITIES (pp. 465-492),

Chapter 16: (9:16)QUADRATIC POLYNOMIALS (pp. 493-.510),and

Chapter 17: (9:17)FUNCTIONS (pp. 511-545)

New concepts on these final hundred pagesemerge directly from prior chapters. Every mo-ment of time saved earlier in the First Course canbe judiciously re-allocated to the sequence hereculminating in the quadratic formula.

34

COMMENTS ON ARTICULATINGSMSG GEOMETRY WITH STUDENTS'

PRIOR SMSG EXPERIENCESThe students should be better prepared for the

study of formal geometry in grade .ten than theywere formerly. They arrive not only with theircustomary first algebra of year before, but alsowith years of informal geometry which has beeninterwoven into the curriculum of the elementaryand the junior high school courses. Since the Birk-holt postulates are utilized in the GEOMETRY,the student's background, which includes realnumbers, meshes well with the geometry, for thereal numbers are used freely in measuring both dis-tance and angles. Some of the GEOMETRY topicsare essentially algebraic, giving the teacher manyopportunities to reinforce and deepen the stu-dent's algebra instead of allowing his algebra tofade into disuse.

With the student's SMSG background the teach-er must be aware not only of the amount of over111Fbut of the depth and length of exposure totopics. A spiral curriculum is considered essentialbut it must not circle in the same place, nor mustit suddenly ascend too rapidly.

Omitting material is rarely the answer to thearticulation problem, for too Often the studenthas unexpected blank spots. The teacher's realiza-tion that the material, as such, is not new, shouldmean, however, (1) a higher level of presentation,(2) perhaps a chmrd emphasis, which takes ad-vantage of the studo..f.'s previous preparatiotl, and,most often, (3) a shortening of allotted time. TheDISC Connelly which seemed impossibly longduring the first year or so can now be completedwith ease in a year or less by students who havestudied the SMSG materials for several years pre-

It is deadly boring for students to "discover"and study topics presented at the same level ingrade 10 that they "discovered" and studied twoorthree years earlier. Teachers who are unfamiliarwith previous texts studied by their students can.easily fall into this trap. In a spiral curriculum itis necessary to repeat materials, but this should bedone with new insights and more depth. (It is im-possible for textbook writers to consider all aspectsof this problem, for some students will study thetext with no previous intuitive geometry.) Withthe background of SMSG texts and with carefulorientation by the teacher, the students are readyfor a formal approach to geometry.

35

Suggested Time Schedule

Part IChapter

1

2345

ApproximateNumber of DaysRecommendedin Commentary

31066

20

Time Recommendedfor Students WithSeveral Veers of

SMSG Background

25-6

4

188

Total Numberof Days

28

12

.I 4f

6 6 7 417 8 8 498 9 7 569 17 17 73

10 6 5 78PART II

11 10 10 8812 15 16 10413 13 14 11814 10 10 12815 5 8 13616 8 8 14417 20 20 164

Given the advantage of several years of SMSGbackground, students should move rapidly

Through the first fot--1. c tapters of transiiion-fromtheir earlier intuitive geometry to the beginningsof formal proof. The teacher may wish to retracefor himself the explorations of informal geometryas cited from SMSG elementary and junior highschool courses in order to identify new abstrac-tions taught at the tenth grade level. It is preciselybecause of dim prior investigations that studentswith SMSG preparations are ready for SMSGGeometry. Here again, comments on articulationare designed to complement, not supplant, theTeacher's Commentary.

SMSG Geometry, Part I

Chapter 1: (10:1)COMMON SENSE AND ORGANIZEDKNOWLEDGE (pp. 1-14)1-1 livo Types of Problems (pp. 1-8)

This is an excellent introduction to the geom-etry if the teacher heeds the commentary and usesThis section as it is intended and not primarily foralgebra review for forgetful students. Articulationproblems occur throughout, the text not only inoverlap of geometric ideas but in overcoming stu-dent weaknesses in algebra.1.2 An Organized Logical Development. of Geom-

etry (pp. 8-14).This section can be considered to be the intro-

duction to the real meat of Chapter IL Even for'lose students with considerable background in

intuitive geometry, this section needs imaginativeexpansion by the teacher, for it sets the stage for

_the_chapters_takillow-,Students-find-it-difficuldistinguish between intuitive geometry and thevery beginning of formal geometry. The logicaldevelopment of the course cannot be overstressedat this point and in the next few chapters. It isvital that students accept the objectives of thecourse if they are to enjoy geometry to the fullest.

The spieit of earlier exposure to geometry inSMSG texts has been to have the student discovergeometric principles and learn correct terminol-ogy and definitions. Little has been said aboutformal proofs in any texts before the first yearalgebra. The algebraic proofs, while fairly rig-orous for the level, are difficult for most studentsand are not sufficient in themselves to show stu-dents the importance and necessity of proof in thestudy of a logical development of Euclidean geom-etry.

Chapter 2: (10:2)SETS, REAL NUMBER AND LINES (pp. 15-511

2-1 Sets (pp. 15-21),2-2 The Real Numbers (pp. 21-27), and2-3 The Absolute Value (pp. 27-30)

Well-prepared students can study these sec-tions in a day or a day and one-half. The ideas and .

vocabulary of sets are quite familiar to studentswith SMSG experience from (E-4:1,5)*, (J-1 :3,4),

References to SMSG texts are in the form (Grade: Chap-ter). (E-4:I, 5) for instance, refers to the fourth grade text;Chapters I and 5.

37

and (9:1). Real numbers are introduced in full in(j -2:1, 6) and (9:1, 5), rationals appear in (E-4:10)

. and (J-1:6), the integers in (E-6:4). Order relation-ships and inequalities recur in (E-4:2), (E-4:10),(E-5:9), (E-6:4), (J-1:3), and 0-2:1); a formal treat-ment of their properties occurs in (9:8). Absolutevalue is defined in (9:5).

2-4 Measurement of Distance (pp. '0 -33)The distance between points as the measure

of a segment is introduced without the concept ofabsolute value in (E-4:9), (E-6:5), and (I-1:7), andwith it in (9:9). Units of measure of distance werestudied in (E-4:9) and (j-1:7). The teacher is urgedto become familiar with those earlier approaclesin order to clarify the new emphasis that thenumber we get as a measure of distance dependsupon the unit of measure.

2-5 A Choice of a Unit of Distance (pp. S3-34)and

-2.6An--Infinite-Ruler-(ppr-35-39)These sections build postulates on such essen-

tially review mawrials as coordinate systems from(E-6:5), (j-2: I), and (9:14).

2-7 The Ruler Placement Postulates. Between-ness, Segments and Rays (pp. 39-48)

Even though most of the ideas will not bestrange to the class, the emphasis given to thisfamiliar material by the teacher at the tenth Gradelevel must be more sophisticated than that of thejunior high school teacher of intuitive geometry.

Lines, rays and segments appear in (E-4:5) andfrequently thereafter. The idea of betweenness isdeveloped in (j-I:4). Under the heading of prop-erty or law several of the postulates have beenstated. Under the heading of postulates, severaltheorems have been stated. These provide excel-lent opportunities for the teachers to discuss whywe develop mathematics differently at variouslevels, emphasizing the maturity of the student asan important' factor.

Contrary to the convention reiterated as re-cently as page 23, number lines on pages 39 and40 have smaller numbers to the right of largerones. With strong students, the teacher might wellrefer back to the rigorous test for y > x, (y > xif y x > 0), suggested on page 219, SMSG FirstCourse in Algebra.

Students should be provided with problemswhere the necessity of an absolute value sign in

38

3) discussing such questions as:a) Can two lines separate a plane into three

regions? (Problem Set 3-3, Problem Mkb) Can three lines in a plane ever separate

the plane into four regions? (ProblemSet 3-8, Problem 14);

c) Problem 2(f) of Set 3-2, "Two parallellines determine a planer

These problems can contribute to the student'sunderstanding if handled informally as intended.

3-2 Theorems in the Form of Hypothesis andConclusion (pp. 60-62)

This section is important to the remainder ofthe course and could use expansion, not con-traction. The student's prior exposures are from0-1:10), where "if-then" statements and their con-verses are mentioned, and (9:7) where "if and onlyif" statements are discussed. Supplementationwith negation and the contrapositrve would pointout that implication is just one of many forms of

_statements_usedin logic.

3-3 Convex Sets (pp. 62-08)This is new content, founded on work with

regions and separations in (J-1:4) and (J-2:11).

Chapter 4: (10:4)ANGLES AND TRIANGLES (pp. 71-95)4-1 The Basic Definitions (pp. 71-77)

Students with SMSG background have workedintermittently with angles and triangles in (E-4:5)and (J-1:4,10). Knowledge of words such as side,vertex, interior, exterior, and notations such asL BAC and iBAC are assumed in subsequentmaterial. Though most of the ideas are familiar,omission is not the solution. The students needreview given in a brief class discussion.

4-2 Remarks on Angles (pp. 77-78)A few brief remarks by the teacher will suffice:"

students will not draw on this content until theystudy trigonometry.

4-3 Measurement of Angles (pp. 79-85)Use of the protractor occurs in (E-5:7) and

U-I:7); however, since most protractors have twoscales instead of the single one printed on pages79-83, a review is essential.

4-4 Perpendicularity, Right Angles and Congru-ences of Angles (pp. 85-92)

40

Categories of angles appear in seventh grade:right, acute and obtuse in (J-1:7); supplementaryand vertical in (J-1:10); Theorem 4-7 is stated in(J-1:10). Perpendicularity of lines, segments, andrays is presented in (J-1:7) and (I-2:4). Congruenceof angles is defined in (E-4:8) and (J-1:7). Workwith triangles appeared in (E-5:4) and (E-6:3).Deductive reasoning in "if-then" form is men-tioned in (J-1:1,10) and in (9:7). Some re-teachingis required to advance undertanding.

Review Problems (pp. 92-95)These exercises provide an opportunity to unify

all the ideas of this chapter before launching intocongruences. However, the teacher must be alertnot to bore students with too much repetition andyet not to take too much for granted in their un-derstanding before their experience of writingproofs. An expert teacher with probing questionscan distinguish between glibness, indicating fa-miliarity with terms but only vague comprehen-

--sion-of-the-meaning,-and real_understandingthough not always well-verbalized. Again, the em-phasis must be on the steady build-up of defini-tions, postulates, and theorems.

Chapter 5: (10:5)CONGRUENCES (pp. 97-154)5-1 The Idea of a Congruence (pp. 97-108)

Congruence of triangles is investigated in(E-4:8), (E-5:4), (J-1:7), and (J-2:4). The techniqueused in each is one of exploration. The idea ofcorrespondence is mentioned in (J-1:4). This sec-tion should be treated as a review, except for thenotation used in writing correspondences betweentriangles.

5.2 Congruence Between Triangles(pp. 109-115) and

5.3 The Basic _Congruence Postulate(pp. 115-117)

The properties of congruent triangles are del-veloped in the fifth grade through tracing geo-metric figures and placing them on others. Condi-tions for congruence, S.S.S., S.A.S and A.S.A., arepostulated in () -2:4). Treat the present sections asreview.

5-4 Writing Your Own Proofs (pp. 117-122)Treat this section as new work; proofs in pre-

vious work in geometry have been informal.

5-5 Overlapping Triangles. Using the Figure inStatements (pp. 123-126)

Treat as new work.

5-6 The Isosceles Triangle Theorem. The AngleBisector Theorem (pp. 127-132)

Properties of an isosceles triangle are exploredin the fourth, fifth, and sixth grades. Theorem5-2 is stated in (E-6:3) and (J-1:10); however,no proof is given. Definitions of scalene, isosceles,and equiangular triangles are covered in (J-1:10).The bisector of an angle is defined in the sixthgrade. Theorem 5-3 is new here. In (J-2:4) thestudent constructs the median of a triangle butit is not named as such.

5-7 The Angle Side Angle Theorem(pp. 132-136)

The A.S.A. theorem was postulated in theeighth grade. This is the first time a formal proofhas been given.

5-8 The Side Side Side Theorem (pp. 137-147)This theorem was postulated in (I-2:4). The

exercises are more difficult than earlier ones andthe fallacy of a circular argument is new. Remindthe students of the fallacious proof they discussedin (9:11, p. 285).

Review Problems (pp. 148-154 )

Time saved earlier in the chapter can be profit-ably spent on these problems.

Review Exercises, Chapters 1 to 5 (pp. 155-158)

The more subtle of these "true or false" ques-tions should be discussed with the class. Misunder-standings and false ideas exist even among goodstudents with a strong SMSG background.

Chapter 6: (10:6)A CLOSER LOOK AT PROOF (pp. 159-188)6-1 How a Deductive System Works (p. 159)

Understanding of what is 'involved in logicalreasoning and setting up a mathematical theorycomes slowly even for well-prepared students.

6.2 Indirect Proof (pp. 160467)Though indirect reasoning has been used in'

justifying properties developed in earlier SMSGtexts and in.proving that V2 is irrational in (9:7)and U-2:6), it has not been labeled as such.

42

6-3 Theorems About Perpendiculars(pp. 167-175)

The idea of perpendicularity and constructionof a perpendicular to a line are taught in (E-6:9)and (J-2:4); however, the proofs here are new.Section 6-3 should be taught as written.

6-4 Introducing Auxiliary Sets into Proofs(pp. 176-181) and

6.5 Betweenness and Separation (pp. 182-188)This is all new material.

Chapter 7: (10:7)GEOMETRIC INEQUALITIES (pp. 189-218)7-1 Making Reasonable Conjectures

(pp. 189-191)The practice of making reasonable conjectures

has been used extensively in SMSG materials,beginning as early as (J-1:1, p. 6) and (J-1:3,p. 77). Relationships between the sides and anglesof a triangle were explored in (E-6:3) and (J-1:10).

7-2 Algebra of Inequalities (pp. 191-192)It is well to review the inequalities of real

numbers (9:8) before considering geometric in-equalities.

7-3 The Basic Inequality Theorems (pp. 193-213)Theorems 7-1 through 7-9 should be treated

as new, although many exercises in the junior highmaterial hint at their existence; for instance, The-orem 7-6 is stated as a property in (J-2:11).

7.4 Altitudes (pp. 214-216)Altitudes of a triangle are investigated in

(E-5:8), redefined and used in (J-1:10) and (J-2:11).Treat this as a review but do not omit.

Chapter 8: (10:8)PERPENDICULAR LINES AND PLANES INSPACE (pp. 219-239)

'8-1 The Basic Definition (pp. 219-221)The properties of lines and planes in space

studied in (J-2:11) form a background for thischapter

8-2 The Basic Theorem (pp. 222-235)Theorem 8.3 was stated as a property in (J-2:11);

otherwise the theorems in this section are new tothe student.

43

8-3 Existence and Uniqueness Theorems (p. 235)This section contains new material.

Chapter 9: (10:9)PARALLEL LINES IN A PLANE (pp. 241-290)9-1 Conditions Which Guarantee Parallelism

. (pp. 241-250)Parallel lines are defined in (J-1:4, 10). The

definition of a transversal is also given in 0-1:10).The relationship between parallel lines and alter-nate inerior angles is discovered in (J' -2:4). Treatas a formalization and extension of previous learn-ing.

9-2 Corresponding Angles (pp. 251-252)Corresponding angles are defined in U-1:10).

Theorem 9-7 is given as a property of parallel linesin 0-1:10).

9-3 The Parallel Postulate (pp. 252-258)Theorem 9.1 irdlsrlinnred-in-U=2:4):-Theorem--

9-9 is stated as a property of .parallel lines in0-1:10). This section is review and extension.

9.4 Triangles (pp. 258-262)Theorem 9-13 is discovered by using known

ps:perties of lines and angles in (J-1;10). Earlier,(E-6:3), it was discovered by mean firing.

9.5 Quadrilaterals in Plane (pp. 263-267)quadrilaterals were introduced in (E-4:5). Op-

posite sides, opposite angles, diagonals, and con-secutive sides have been defined. A trapezoid isdefined in 0-2:11). A parallelogram is defined inU-1:10) and many of its properties discussed. The-orem 9-14 was proven as an exercise in 0 -2:4).Theorems 9-15, 9-16, and 9.17 are given as proper-ties of a parallelogram in (J-1:10). The distancebetween two parallel lines is defined in 0-1:10).Theorem 9-18 is used in an exercise of 0-2:4).Theorems 9-19 through 9.22 are new.

9-6 Rhombus, Rectangle and Square(pp. 268-274)

The definitions of rhombus, rectangle, andware hive been given in previous SMSG texts

(E-5:8), (J-1:8), and (J-2:11). Theorems9.23 through 9-25 are new to students.

9-7 Transversals to Many Parallel Lines(pp. 275-283)

Concurrent lines arf,,defined in 0 -2:4). Theo-

44

rem 9-27 is discovered through an exercise in(J -2:4). Meat as a review and extension of con-cepts.

Chapter 10: (10:10)PARALLELS IN SPACE (pp. 292-315)10-1 Parallel Planes (pp. 292-299)

Parallel planes are discussed in (E-4:5), de-fined in (J -1:4) and (1-2:11). Theorem 10-2 appearsin an exercise and Theorem 10-5 is stated as aproperty in 0-2:11). This section should betreated as an extension of previous concepts.

10.2 Dihedral Angles, Perpendicular Planes(pp. 299-306) and

10.3 Projections (pp. 306-312)This is new material.

SMSG Geometry,- Part -II

Chapter 11: 00:11)AREAS OF POLYGONAL REGIONS(pp. 317-357)11-1 Polygonal Regions (pp. 317-527)

Polygonal regions are defined in (E4:5) indiscussion of triangles. iliangular and quadri-lateral in,regions are .again met (E-5:8). Spaceregions are defined in (E-6:7) and (J-2:10). Postulate 20 has been used in (E-5:9), (J-1:7); andU-2:11). Problem 8 is discussed in (J-2:10) and-2:13). This section unifies and extends such

material.

11.2 Areas of 'Mangles aztAi Quarilaterals(pp. 3211-338)

Area of a triangle was introduced in 7-5:3)and met again in (J -1:10). Theorem 11-2 is used'in (E-5:8) and (J-1:10) and stated as a formula inij-2:11). Theorem 11.3 is stated in (J-1:10) and-2:11). Theorem 11.4 is stated and written as a

ormula in (J-2:11). Meat as review leading toextension.

11.3 The Pythagorean Theorem (pp. 339-353)The Pythagorean theorem is stated and- two=

proofs are given an (1-2:4). An extremely simplenonalgebralc proof, adapted to an overhead pro-jector, appears in SMSG Introduction to Second-ary School Mathematics, Volume 2, Part II, Chip-

.

45

er 19. Special triangles (Theorems 11-9 and 11-10)treated in (E-6:3), Here these topics are impli-

ed, formalized.

pier 12: (10:12)

IMILAR1TY (pp. 359-403)

2-1 The Idea of a Similarity (pp. 359-364)Similar triangles, not identified as such, were

to introduce ratio and proportion in (J-1:91.roportions were then used in the balance of the

pter to solve percent problems. The ninth,ide text contains but brief references to ratio

nd proportion (9:9, 15). The definition of geo-etnc mean is new to students.

12-2 Similarities between Teiang les (pp. 364-367)Similar triangles are defined in (J-2:9). Treatis section as an extension of previous concepts.

iMilitityTheorrMs(pp. 367491)

The theorems are new; they draw on exercisesn 0-2:9) discovering the principles of Theorems2-3, 12-4, and 12-5.

12-4 Similarities in Right 'Mangles (pp. 391494)and

12-5 Areas of Similar Triangles (pp. 395-400)All new material.

Chapter 13 (10;13)CIRCLES AND SPHERES (pp. 409-460)13.1 Basic Definitions (pp. 409-412)

The student should have a broad backgroundfor the topics given new formalization here.Circles named by their centers and determinedby their radii are studied in (E-4:5), (E-6:9), andU-1:11). The arc and its center are presentedin (E-5:4), (E-6:9), and (J-I:11). In (E-6:9) and(j -1:11) can be found defirltions of chord, diam-eter, and central angle. The semicircle is includedin 0-1:11). Theorem 13-1 appears in (J-2:12).

13.2 Tangent Lines. The Fundamental Theoremfor Circles (pp. 412-422)

Tangents are mentioned in (J-1:11) and muchpound work has been laid for Theorem 13-2.Some re-teaching is essential.

13-3 Tangent Planes. The Fundamental Theoremfor Spheres (pp. 423-438)

Spheres have not been given extensive treat-ment, although they were mentioned in (E-4:8).(J-2:12), often omitted from the eighth grade cur-riculum, introduced great circles and tangentplanes. This section (13-3) may be discussed intui-tively, since it parallels very closely the previoussection, (13-2).

13-4 Arcs of Circles (pp. 429-447)Though the students have had some back-

ground from (E-6:9) and (J-1:11), most of thissection will be new to them.

13-5 Lengths of Tangents and Secant Segments(pp. 448-456)

This material is new.

Chapter 14: (10:14)

CHARACTERIZATIONS OF SEGMENTS.CONSTRUCTIONS (pp. 461-504)

14-1 Characterization of Sets (pp. 461-463)Sets have been an unifying concept throughout

SMSG texts. This is a new approach.

14-2 Basic Characterizations. ConcurrenceTheorems (pp. 464-473)

In (E-6:9) and (J-2:4) siudents learned to dosome, compass and straightedge constructions.Foundation for Theorem 14-4 was laid in an exer-cise of U-2:4).

14-3 Itpersection of Sets (pp. 473-475)The language and ideas are familiar but the

approach here is new.

14-4 Construction with Straightedge andCompass (pp. 475-477)

The student had some experience with con-structions in (E-6:9) and (J-2:4).

14-5 Elementary Constructions (pp. 477-489)Constructions 14-7, 14-8, and 14-8-1 appear

in (J-2:4). Construction 14-9 appears in both(E-6:9) and (J-2:4). The teacher needs to be awareof this background and present the ny.terial tak-ing advantage of recent months of formal geom-etry.

14-6 Inscribed and Circumscribed Circles(pp. 490-493)

This is not essentially new material but theevelopment is.

4-7 The Impossible Construction Problem's ofAntiquity (pp. 493-502)

This material is new for most students, andsually promotes such a lively discussion thatxtra time is needed.

evieW Problems (pp. 503-504)There are few problems here; if there is time,

he student should be provided with additionalview problems before proceeding.

hapter 15: (10:15)REAS OF CIRCLES AND SECTORS

pp. 505-531)5-1 Polygons (pp. 505-509) and5-2 Regular Polygons (pp. 510-515)This section builds on the formula for the

urn of the angles of an 71-sided polygon fromE-6:3) and the regular polygons studied in-2:11).

5-3 The Circumference of a Circle. TheNumber ir (pp. 516-519) and

-5-4 Area of a Circle (pp. 520-524)In (J-1:11) circular regions are discussed and

here is some attempt at that point to arrive at thealue 7r. through measurement, but the formulaor circumference of a circle is essentially assumed.n U-1:10) the formula for the area is simplytated, not rationalized as the elaboration on-2:11) is here.

15-5 Lengths of Arcs. Areas of Sectors(pp. 525-529)

Length of an arc and area of a sectorussed in (E-2:11) as part of investigations If the

lateral area of a cone, are extended in this section.

Review Problems (pp 530-531)In reviewing the chapter, supplementary prob-

ems like number seven and number seventeen addinterest. .

Chapter 16: (10:16)VOLUMES OF SOLIDS (pp. 533-565)16-1 Prisms (pp. 533-539),16-2 Pyramids (pp. 540-545),

16-3 Volumes of Prisms and Pyramids,Calvalieri's Principle (pp. 516-553),

16-4 Cylinders and Cones (pp. 553-559), and16-5 Spheres: Volume and Area (pp. 559-563)

This material is not totally new to the students. Prisms, altitude, cross-section, lateral edgeface, and surface were studied in (E-4:8), (E-6:7)and (J-2:11); parallelepipeds, in (J-2:11). Pyramidhave been met i, (E-4:8), (E-6:7), :.end (j-2:11).

Formulas in Theorems 16-7,16-9, and 16-10 arstated and used in (E-6:7). Theorem 16-7 and Theorem 16-10 are in (J-2:11). Theorem 16-15 is presented in (J-2:11). The volume of a circular cyiinder is discussed in (J-1:11). The volume and surface area of a sphere occur in (J-2:12). All of thiintroductory work was done without proofs. Treathe present sections as extensions on the familiaideas.

Chapter 17: (10:17)

PLANE COORDINATE GEOMETRY(pp. 567-629)17-1 Introduction (p. 567)

It is to be hoped that teachers have takeenough advantage of students' background iintuitive geometry to reach this important finachapter before the close of school. The entirchapter is really too brief an introduction, anwould. he enhanced by supplementation froSMSG Geometry with Coordinates, Part H.

17-2 Coordinate Systems in a Plane (pp. 567-572

17-3 How to Plot Points on Graph Paper(pp. 572-575), and

17-4 The Slope of a Non-Vertical Line(pp. 576-583)

Essential, review. of (E-6:5), (J-2:1), (J-2:9),(9:14), and (9:15).

17-5 Parallel and Perpendicular Lines(pp. 583-587)

Theorem 17 -2, appears in (9:14) but Theorem17-3 is new.

17-6 The Distance Formula (pp. 588-591)This is more formalized than any presentation

previously.

17-7 The Mid-Point Formula (pp. 592-595)Students find the formula intuitively simple

49

to understand, but the proof, which is new, is moredifficult for them.

17-8 Proofs of Geometric Theorems (pp. 595-.600)This is new and difficult, and requires more

time,than it appears. Expansion is desirable.

17-9 The Graph of a Condition (pp. 600-604)The students have been well prepared for this

section in the graphing of the junior high schooltexts and in (9:14, 15). Some of the more difficultproblems from the ninth grade text might well beused as supplementary work here.

17-10 How to Describe a Line by an Equation(pp. 604-611)

This review of (9:14) is essential here.

17-11 Various Forms of the Equation of a Line(pp. 611-613)

Theorem 17-8 appears in (9:14), but the stu-dents need this new explanation.

17-12 The General Form of the Equation of aLine (pp. 613-616)

Though Theorem 17-10 appears in (9:14), theteacher must not assume that the students remem-ber it or can use it

17-13 Intersection of Lines (pp. 617-621)The intersection of lines found by graphing

and by the algebraic solutions of two linear equa-tions appear in (9:15). The vocabulary here. is'slightly different.

17-14 Circles (pp. 621-628)This is new material.

50


Newsletter No. 28

January 1968

ARTICULATION OFCONTENT OF SMSG TEXTSGRADES 1-3 and__GRADE 4.


E:- 'Beg leTO ERIC AND ORGANIZATIONS OPERATING .

UNDER AGREEMENTS WITH THE NATIONAL IN-STITUTE OF EDUCATION. FURTHER REPRO-DUCTION OUTSIDE THE ERIC SYSTEM RE-WIRES PERMISSION OF THE COPYRIGHTOWNER."

C 1968 by The Boani of Trustees of theLeland Stanford Junior UniversityAll rights reservedPrinted in the United States of America

ARTICULATION OF CONTENTOF SMSG TEXTS

FOR GRADES 1-3 and GRADE 4

In the development of the sample texts for pu-pils and commentaries for teachers which com-prise SMSG's Mathematics for the ElementarySchool, the materials for grades 4-6 preceded thematerials for grades K-S. Within each of thesetwo levels-intermediate grades and primarygrades-the content sequence was sufficiently wellarticulated. This is not as true, however, for thecontent articulation between Books 3 and 4. Andfor the following reason:

The texts for the intermediate grades necessar-ily were written for pupils who had no compar-able kind of background previously in the primary

ides. Thus, a portion of the material for grade4 presented to pupils a new view of some of themathematical content that had been studied froma traditional point of view and in a narrowly lim-ited way during earlier grades..

When in due course of time text materials weredeveloped from a contemporary point of view for

ades K-3, a certain lack of articulation betweenBooks 3 and 4, in particular, was inevitable. It waseemed better to face this situation honestly than

to try to avoid it by having the content of Book 4lace an artificial delimitation upon the contento be included in Books K-3.

This present report, prepared by J. E Weaver'th the assistance of Carole Greenes and Priscillareer, is similar in its intent to the report on

'Articulation of Content of SMSG Texts, Grades-10" (Newsletter No. 25; December-1966):In this instance, however, we concentrate on the

rticulation of text content for grades 3 and 4,'th some reference to grades 1 and 2 whereverppropriate and helpful. The material-resentedn this report should facilitate greatly the transi-'on from the use of SMSG texts in grades 1-3 toe use of SMSG texts in grades 4-6, with empha-

is upon a smoother articulation of content forades 4 following the sequence for grades 1-3.In this report, references to SMSG texts appear

n the following forms. 3. IV-5 (grade, chapter,ction), or 3: IV (grade, chapter),.or IV-5 (Chap-

section). Page numbers, where indicated, refer,o the Teacher Commentary, not the Student Tekt.,

GRADE 4Chapter 1CONCEPT OF SETS (pp. 1-50)

All conceptual ideas in this ,chapter have beenintrodioced and used on numerous occasions inBooks 1,2 and 3. This material is essentially a re-view with these two extensions:I-5. Union of Sets (pp. 29-35)

The symbol for set unionsU, is new content.1-6. Intersection of Sets (pp. 36-42)

The symbol for set intersection, n, is new con-tent. Note that the concepts of union and inter-section are not new.

Chapter 2NUMERATION (pp. 51-109)II-1. Grouping in Base Five (pp. 60-68)11-2. Base-Five Notation (pp. 69-75)

11-3. Grouping and Notation in Other Bases(pp. 76-81)

This is the first time that explicit considerationhas been given to a scheme of place-value notationhiving a base other than ten. This should be de-',veloped as new content. It is not intended that:-pupils become skillful in this Work; rather, that.they gain some appreciation of base and its sig-,nificance in connection with place-value notation.11-4. 'A Number May Have Several Names (pp.,

82-85)

Numbers (pp: 86=91)

11-6. Extending Ideas of the Decimal System(pp. 92-98)

11-7. Order Relations on the Number Line(pp. 99-105)

11-8. Some New Symbols (pp 106-108)

11-9. Just for Fun (p. 109)All concepts and symbolism used in these sec

tions have been developed progressively duringthe course of Books 1, 2 and 3. Basically this, is: review material, except that some extensions aremade beyond4-place numerals considered in BOok3, to 5- and 6-place nunierals here in grade 4.

Chapter 3PROPERTIES AND TECHNIQUES OFADDITION AND SUBTRACTION I(pp. 111-228)

III-1. Addition and Subtraction (pp. 117-123)This essentially reviews the interpretation of

addition in relation to the union of disjoint sets,.and should be a.very familiar idea to pupils.111-2. Addition and the Number Line

(pp 124-128)An interpretation of addition in relation to the

number line was made very explicit in (2:II-1),but received less detailed consideration in Book 3.Although this is essentially review, this interpreta-tion of addition is an important one and should .

not be slighted.111-3. Addition and Subtraction as Operations

(pp 129-133)This is an overview of a higher level interpre-

tation of addition and subtraction (and multipli-cation and division) as operations. Treat the es-sential nature of this idea as new material.111-4. True Mathematical Sentences (pp. 134-142)

Mathematical sentences, and number sentencesin particular, should be familiar to pupils fromBooks I, 2 and 3. However, this section makesmuch more explicit than before the fact that asentence may be true or it may be false. Treat thematerial accordingly.111-5. Thinking About Addition Facts

App._143,150)This is clearly review material.

111-6. The Commutative Property for Addition(pp 151-156)

This property has been developed intuitively in.preceding grades. This is the first time, however,.that the property has been made this explicit and:given 'a name (commutative).111-7. .Thinking About Subtraction Facts

(pp. 157-166)Although subtraction facts are familiar to pu-.

.pils, here subti-action is interpreted explicitly interms of finding an unknown addend. Althoughthis interpretation has been implied in earlier

grades, you will need to deal carefully with thisinterpretation in this section.111-8. Mathematical Sentences Using the Number

Line (pp. 167-169)111-9. Number Line Pictures and Mathematical

Sentences (pp. 170-172)These sections review briefly section 111-2 of this

chapter and extend the work to interpret subtrac-tion in relation to the number line..III-10. More Mathematical Sentences

(pp. 173-178)111-11. Using Mathematical Sentences in Problem

Solving (pp. 179-189)These related sections provide higher level in-

terpretations of addition and subtraction, as havebeen developed during the course of this chapter.The emphasis is upon addition as an operation forfinding the sum when two addends are known,and upon subtraction as an operation for findingthe unknown addend when a sum and one addendare known. The application of these interpreta-

, tions, and of mathematical sentences, to problemsolving situations should be treated carefully andin some depth.III-12. Doing and Undoing Addition and

Subtraction (pp. 190-197)This is a review of material introduced prior' to

this grade, but the treatment is more explicit thanbefore.III-13. More About Addition and Subtraction of .

Whole Numbers (pp. 198-206)Although- the ideas of this section have

glimpsed implicitly in earlier work, this materialshould be handled essentially as new content: anexplicit recognition of the closure property foraddition within the set of whole numbers (and thelack of this property in the case of subtractionwithin the set of whole numbers).III-14. More Problem Solving (pp. 207-212)

This extends the work of section III-11 of this`?chapter.

.111-15. The Associative Property for Addition(pp. 213-220)

Although this property has been used in an im-plicit way in preceding grades, this is the first ex-

plicit statement of the property. The materialshould be treated accordingly.111-16. Review (pp. 220-228)

Chapter4PROPERTIES OF MULTIPLICATIONAND DIVISION (pp. 229-386)

These operatiOns, and properties associatedwith them, have been developed extensively inBooks 1,2 and 3:

1: VIII Arrays and Multiplication2: VIII Arrays and Multiplication2: IX Division and Rational Numbers3: IV Arrays and Multiplication3: VII Multiplication, Quotients, and

Division3: IX DivisionIn light of this extensive background, there are

very few instances in which the material of thischapter is not strictly review and may be treatedaccordingly. In addition to a more explicit interpretation of division as an operation for find-ing. a missing factor when a product and onefactor are known, the following sections should benoted:IV-6. Using the Number Line (pp. 287-289)

. This interpretation of multiplication may betreated as new content.IV-10. Closure (pp. 313-316)

This makes more explicit the closure propertyas it does or does not apply for the operations of .

multiplication and division within the set ofwhole numbers.IV -12. Distributive Property of Division Over

Addition (pp. 333-337)Treat this as new content.

IV-15. Partitioning Sets (pp. 353-356)

IV-16. Making a Record of Our Thinking(pp. 357-362)

The ideas presented in these two sections arenot new. However, pupils encountered two thingsin connection with tkeir division work' in Book3 that are not.in any way reflected in the division

work in Grade 4:(1) an expression such as "32 4" also was sym-

bolized as4

Each expression was read in the

same way: "32 divided by 4:' (See 3:V111)(2) Because of more extensive work with rational

numbers in Book 3, the quotient for a division2such as 32 ÷ 5 was expressed as 6 +-3; that is,

32 5 = 6 + 2. Pupils who have used Book 3 are5

more familiar with this way of handling "re-mainders" than with the way used here -iii .whichthe following sentence is associated with "32 ÷-5": 32 = (6 x 5) + 2. This latter form is appro-priate here, but you also may wish to havechildren recall the way in which they handledremainders in the Book 3 approach. (See 3:IX)

Chapter 5SETS OF POINTS (pp. 387 -491).

Extensive work with non-metric geometry haspreceded this chapter; e.g.,

1: V Recognizing Geometric Figures2: III Sets of Points2: VII Congruence of Angles and Triangles3: I Sets of PointsThe material here in Chapter 5 can, for the

most part, be developed as a review of ideas withwhich children already have some familiarity.However, the approach and organization aresomewhat different at times from that used in.Books 1, 2 and 3. This has ari obvious motiva:tional advantage.

The following sections of this chapter are theones that involve either new material or an ex-plicit extension or higher level interpretation ofmaterial:V-7. Planes (pp. 430-436)V-8. Lines and Planes (pp. 437-444)V-9. Intersection of Lines and Planes

(pp. 445-452)

V-12. Circles (pp. 465-472)V-13. Regions in a Plane (pp. 473-477)V-15. Angles of a Triangle (pp. 486-491)

Chapter 6PROPERTIES AND TECHNIQUESOF ADDITION ANDSUBTRACTION II (pp. 493-563)

This chapter is devoted to the development offurther skill in computing, using addition andsubtraction, and to applications of such skill. Thetechniques presented, and the principles uponwhich they are based, are no different than thoseinvolved- in (3: V). Now, however, numbersgreater than 999 are used as addends = extendingto those named by 4- and 5-place numerals.

The only relatively new or novel content is inthese sections:VI-11. If-Then Thinking (pp. 547-550)

This makes explicit a kind of relational think-ing that pupils have used informally and implic-itly before.VI-13. Enric.hment (pp. 558-562)

This extends, on an optional basis, the interpre-tation of a mathematical operation.

Chapter 7TECHNIQUES OF -MULTIPLICATIONAND DIVISION (pp. 575-692)

In the preceding year, in addition to "basicfacts': pupils should have developed skill in mul-tiplication and division to the following extent:

Multiplying and dividing with multiples of 10(3: VII)Multiplying numbers between 10 and 20 bynumbers less than 10 (3: VII)"Division with remainders' as delimited by useof basic facts (3: IX)Also, pupils should be familiar with an algo-

rithm for the division process and with theseforms of notation associated, with division:

23 . 23 223+,7 =and= 3 +7 7 7

;Neither of these forms is used in Book 4, how-evir-]

It also is true that pupils in Book 3 (3: IX) deal'with situations in which they indicate whether,',

tor8

instance, of 48 is greater than equal to, or. ,

4Of 42. [This kind of work is not

considered in Book 4.]The following sections of this chapter include

extension of skill in computing with multiplica-tion and division, using lumbers greater thanthose used in earlier years; consequently, a degreeof "new work" is present in each of these sections:

VII-2. Multiplying by Multiples of Ten(pp. 582-586)

VII-3. Multiplying by Multiples of One Hundred(pp. 587-591)

VII-4. More About Multiplying (pp. 592-598)

VII -5. Multiplying Larger Numbers(pp. 599-604)

VII-6. A Shorter Method of Multiplying(pp. 605-608)

VII-7. Multiply Numbers Less Than 100 byMultiples of 10 (pp. 609-613) .

VII-8. Finding Products of Any. Two NumbersGreater Than Ten (and Less Than OneHundred)- (pp. 614-623)

VII-9. Using Multiplication in Problem Solvin(pp. 624-627)

VII -10. Finding Unknown Factors (pp. 628-637)

VH-11. A Way of Dividing Two Numbers(pp. 638-641)

VII -12. More About Dividing Two Numbers(pp. 642-645) .

VII-13. Using Division in Problem Solving(pp. 646-651)

VII-14. Becoming More Skillful in DividingNumbers (pp. 652-659)

VH-15. Finding Quotients and Remainders(pp. .660 -666)

VH-16. Reviewing and Extending (pp. 667-692

As undoubtedly is evident, this is a "meaty!chapter. It often is handled most effectively whespread over an extended period of time, intespersed with geometric content.

Chapter 8RECOGNITION OF COMMONGEOMETRIC FIGURES (pp. 693-766)

In light of the extent of work in non-metricgeometry in Books 1,2 and 5 that was mentionedearlier, much of this material is an extension andrefinement of that work and is not "new" in astrict sense. It would be well to note particularlythe following things: however.VIII-1. Review the Triangle and Quadrilateral

(pp. 703-707)Note the use of "vertex" and "vertices" as ap-

plied to polygons.VIII-3. Isosceles and Equilateral Triangles

(pp. 713-718)Note the construction involving compass and

straightedge.

III-4. Right Angles (pp. 718-723)Note the comparison of angle sizes that is in-uded in this section.111-5. Rectangles and Squares (pp. 724-727)Note the explicit statements of relationship be-

ween rectangles and squares.III-6. Surfaces (pp. 728-766)YoL may develop this as essentially "naterial.

apter 9INEAR. MEASUREMENT (pp. 780-841);,As with several other aspects of content, pupilsave had considerable experience with lineareasurement in. Books 1, 2, and 3:1: X Linear Measurement2: V Linear Measurement3: VI --Length and AreaThis chapter can be developed principally'thin a review context, with no new mathemati-

al ideas being introduced. But note in particulare following things:

X-4. Measuring a Segment (pp 792-794)Note the clear distinction that is made in con

ection with these expressions or terms: "unit'ofLeasure; "a measure as a number," and "length."

IX-7. Using Standard Units of Length(pp. 800-807)

Note the use of the centimeter as a unit.Although all of this work should appear famil.

iar to pupils, it is presented at a somewhat higherlevel of approach than was true in preceding'.years.

. Finally, notice that in .(3: VI) pupils were in-troduced to the concept of the area- of a planeregion, but such work is not included in the textcontent for Grade 4. Be certain in connectionwith sections (IX-1I) and (IX-12) of this chapterthat pupils do not confuse the concept of areawith that of perimeter.

Chapter 10CONCEPT OF RATIONAL NUMBERS(pp. 845-934)

First note the frequency with which ideas re-garding rational numbers have been developedin Books I, 2 and 3:

I: IX Partitions and Rational Numbers2: IX Division and Rational Numbers3: VIII Rational NumbersIn this preceding work the non-negative ra-

tional numbers have been associated with: (I)partitioning regions into congruent regions, (2)partitioning sets into equivalent sets, (3) parti-tioning segments into congruent segments, (4),points on the number line, and (5) the operation.of division. Clearly, this background is more thanadequate to make the most of this chapter (4: X)essentially review in nature.

There is one section of this chapter, however,for which a special note is in order:X-6. A New Kind of Name (pp. 911-917)

Previously in (3: IX) pupils were introduced3 Ito statements of equality such as: T. 4 -FT; but

there was no explicit attempt made to shorten13 1this to the following form:-- 4-3-. This abbre-

viated form is now used in this section, and anexcellent background for its interpretation hasbeen given through the work in (3: IX), withwhich pupils should be familiar.

12

CONCLUDING NOTES

In the preceding outline we frequently haveindicated that topics in Grade 4 can be developedwithin a "review context" because of the ratherextensive treatment which preceded in Books 1,2 or 3.

You will find, however, that the level of pres-entation and the level of approach are appropri-ately higher in Grade 4 than for Books 1, 2 and 3.This can be a distinct advantage in that familiarcontent is not simply "rehashed" for pupils, butis truly re-viewed: viewed in a light which bringscontent into a sharper focus than was true inearlier years. Do all that you can to help pupilssense this fact as they work anew with ideas that,in essence, are familiar to them.

Pupils will bring to Grade 4 an uncommonunderstanding of what we call the whole-numberline in light of their work in Book 3 with (3: III):Describing Points as Numbers. You can use thisbackground to good advantage by making morefrequent use of the number line that is indicatedspecifically in the text material for Grade 4.

13


Newsletter No. 33September 1970

MATHEMATICSFOR DISADVANTAGEDAND LOW ACHIEVINGSTUDENTS

Early in its history, SMSG demonstrated thatt was not concerned solely with above averageollege-bound students. The first junior andnior high school SMSG texts were designed forllege-capable students, but even before theseere finished work had been started on a revisionf the texts for grades 7, 8, and 9, which would beore useful for average and below average stu-

ems than the original versions. These texts, IN-RODUCTION TO SECONDARY SCHOOLATHEMATICS, Volumes 1 and 2, and IN-RODUCTION TO ALGEBRA, were finished1962.

a a a

During 1963, arrangements were made for audy to be carried out during the following twocademic years in which these texts were used byu_ dents between the 25th and the 50th percentilegeneral ability. This study demonstrated that

ch students could learn about as much mathe-atics as above average students provided theyere given more time. Further information onis study is contained in SMSG Report No. 5.

a a a

In 1964, in order to obtain comments and sug-estions from the mathematical community,MSG convened a conference to discuss all aspectsf mathematics education for below averagechievers. A report of this conference was Wa-shed.

a a *

As a result of this conference, SMSG took ono new activities. One was concerned with pri-ary school mathematics for disadvantaged chil-

n and led to a revision of the SMSG materialsr kindergarten and grade one.Two assumptions were made in revising theginal SMSG books to make them more useful

r children with backgrounds that differ signifi-tly from what is usually thought of as "middle

d/or upper class."1. These children are entitled to a mathematics

curriculum which is mathematically soundand properly sequenced from both a mathe-matical and pedagogical point-of-view.

2. They-can learn from such a curriculum if

thematerial is presented in a way whictakes into account the presence or absencof the special skills, attitudes, and behaviowhich the school seems to demand as a prrequisite for academic success.

Since the existing books had been revised ilight of teacher reports during a year try-out, onsite observations and the recommendations of consultants, the revised editions met the requiremenof the first assumption.

To find out if the material was appropriate fodisadvantaged children and if not, how it mighbe modified for maximum benefit, SMSG set ucenters in Boston, Chicago, Detroit, Miami, Oaland (California) and Washington, D. C. Sevekindergarten and eight first grade teachers antheir classes were involved. The designation ofparticular school in the disadvantaged rorea oeach city, as well as the selection of teachers antheir classes were made by the local school syste

The fifteen teachers were provided with all thavailable educational and play materials suitablfor their classes as well as the revised SMSG textEach teacher had available the services of a mathmatics consultant and a psychologist at regulintervals.

. .The teachers made weekly reports describmand evaluating their daily mathematics lessonand following the progress of individual studentThey prepared reports on each chapter of thSMSG books as they completed the material. Thteachers also met, as a group, with committmembers of SMSG four times during the schyear to discuss progress, report difficulties and trecommend modifications of the existing SMStext materials. All classes were visited at least oneby a member of the SMSG staff at Stanford Uriversity.

As a result of the teachers' reports, the groumeetings, and the on-site observations and recomendations of mathematics consultants and pschologists, the existing SMSG kindergarten anfirst grade texts were rewritten. They are nopublished under the same title followed by "Scoal Edition."

The Special Edition of the kindergarten booreflects teacher concern that many different actities and types of material be readily at hand tintroduce and reinforce the concepts which arconsidered preparatory to first grade work. Teacers are encouraged to look upon this period as onwhich is exploratory and developmental. Masteof concepts by every child is NOT implied.

The Special Edition of grade one begins -with areview of the concepts presented in kindergartenbut provides a complete teaching procedure sothat the child without kindergarten experience isnot penalized. Many more student pages are avail-able where teachers have indicated a need for aslower pace in the development of a particularconcept. In some cases the order of presentationwas altered. For example, teachers found it easierto teach the numbers 11-19 after ten and the

ultiples of ten less than one hundred had beendeveloped rather than stress the numbers 20-99and then return to the teens. Those pages which

e to be used in the teaching of a lesson are soarked. In the teacher edition many activitieshich stress the use of manipulative materials are

suggested before a given page in the book is to bensidered, If the concept is a difficult one, suchthe idea of "fewer than," for ALL children, the

teacher is alerted to this fact in order to avoidrustration at the lack of immediate results. Withhese children the pace will be slower and theange of achievement is apt to be greater than inmiddle class heterogeneous group. Consequent-

y, teachers are made aware of situations whichill require a great deal of development and theecessity of making additional worksheets to fit

he individual differences in the class. Unhappilyor teachers, there is NO book of any kind orescription which has enough pages of varyingifficulty to fit the range of needs in any one givenass.For the benefit of teachers of primary school

isadvantaged children, a new volume, No. 13,as added to the Studies in Mathematics series.his book was written at the request of teachers

nd consultants who indicated a need for a bookevoted to the mathematics and pedagogy of antemporary instructional program that empha-

izes conceptual learning.. The beginning chapteriscusses the factors which contribute to being dis-dvantaged, a description of such children ande implications for teaching. The followingventeen chapters introduce and provide a pro-essive development of significant ideas in the-3 program. In each chapter the exposition of a

oncept and the related development of appro-date skills is followed by a section called "Ap-lications for Teaching." This unusual feature isn important one since the mathematics dictateshe pedagogy. The reader is told what ideas, lan-

age, etc. may prove difficult for disadvantagedhildren and provides suggestions which other

teachers have found helpful. Before the usual list-ing of new vocabulary and exercises for each chap-ter is a small section entitled "Question." Thequestion or questions considered here are the onesmost frequently raised by teachers of disadvan-taged children when working with the mathemat-ics consultant. They highlight the points whichseem to cause teachers the greatest amount of mis-understanding and erroneous interpretation.

The appendices deal with (1) the scope and or-ganization of the K-3 program, (2) .the usefulnessand difficulties of the language of mathematicand (3) a report of the study of disadvantagechildren which supplied the information needeto write the Special Editions and this volume.

SMSG Reports 2 and 4 provide further information on these activities.

A by-product of these activities was a bookiemeant for nursery school teachers who are working with disadvantaged children. A number omathematical ideas which are normally explorin kindergarten or the first grade are described.For each of these, a number .of nursery schooactivities are suggested which should facilitate thstudent's later development of these topics ikindergarten or first grade. All the activities suggested were tried out with disadvantaged nurserschool children and found to be feasible.

A second SMSG activity undertaken as a rsuit of.,the 1964 conference is now culminatinin the publication of a junior high school mathematics program SECONDARY SCHOOMATHEMATICS SPECIAL EDITION designed for students whose mathematics achiement in elementary school was very low. The firsnine chapters, more than enough for one schooyear, were made available in September 1970 ananother year's worth of material will be madavailable the following September.

The mathematical content of this junior higschool program is derived from the new SMSSecondary School Mathematics Program, described originally in SMSG Newsletter No. 24 anagain more briefly in Newsletter No. 30., Howeverthe format of this special edition is a deciddeparture from that of the usual classroom textbook.

This change of format comes as a result ofsmall experiment, conducted over a three-yeperiod, with junior high school very low achiever

6

n mathematics. Details of this experiment will beound in SMSG Reports 6 and 7. The heart of

is experiment was the development of materialshat would relieve the student from the burdensf computation, as much as possible, and concen-ate on mathematical concepts and relationships.ables were provided to enable the student toompute whenever the content of the material soctated. A second departure from the norm was

hat students were not provided with textbooksbut were issued daily. worksheets on which theesson for the day was printed and which con-lined ample space to do any of the requiredork. Worksheets were placed in a binder which,

n most cases, was kept in the classroom. Thereere two factors that prompted this approach.irst, because of these students' apparent imma-urity and lack of organization, a textbook be-

me a handicap to them. Simply keeping track ofis physical location appeared to be beyond theipabilities of most of the students. Secondly, thispproach seemed to produce the more liositiveffect of having the students consider what theyad accomplished; rather than projecting whatey had yet to do. These two aspects of the ex-riment proved to be extremely successful both

rom the point of view of the student and theeacher. The students included in this experimentearned about as much mathematics as similartudents in control classes; however, there werearked attitude changes among the experimental

tudents which did not appear among; the controltudents and these changes were deemed to beesirable ones. In addition, discipline problemsere markedly lower in the experimental classes

han in the control classes. SMSG has retainedheie features in the development of this new ma-ids'.

A, group of-writers prepared experimental ver-ons of nine chapters during the summer of 1969.tiring the 1969-1970 school year, sixteen I venthade classes, taught by fourteen teachers in

liven different schools, tried out the experimen-t chapters. All the students were low achievers

n mathematics. Several classes consisted of Blackudents and two of Mexican-American students.ost of the classes finished eight chapters. Theactions of the students, the parents, and the

eachers were extremely favorable. Teacher evaludons for each chapter were systematically colected and these evaluations served as a basis for

e revision of the nine chapters during the sum-er of 1970.

Throughout the trial period participatinteachers attended biweekly seminars in which thmaterials and teaching problems were discussedProblems which arose were carefully noted anways and means of counteracting them have beeincorporated in a short teachers commentary. Sugestions for handling the material have also bee

included in the commentary.

The following are chapter headings of the material available:

Revised Version1. Flow Charts2. Structuring Space3. Functions4. Number Theory5. The Integers6. Rational Numbers7. Probability8. Equations9. Congruence

It should be noted that although this materiais presented to the student in the form of worksheets, it is not, in the ordinary sense, a workbooTopics in each chapter are sequentially develolesson by lesson. Teacher-led class discussion exercises are carefully programmed to lead the studento successful experiences in the exercise sets. Evereffort has been made to construct lessons that prevent failure. The, quantity of reading and threading level have been reduced to a minimumAt the end of each chapter there are included(1) a cumulative "self-test," which enables thstudent, on his own, to determine how he is pgressing; (2) a practice test, which in essence, tellthe student what he is expected to know; an(3) a chapter test which is administered by thteacher.

As the cost of producing individual worksheetcommercially proved to be prohibitive, this mameal will come to the teacher in bound volumesEach chapter will consist of a number of lessonof one or more pages. Each page will be pertrated for easy removal. These pages can thenreproduced in quantities sufficient for the class bmeans of spirit master units. Although this reprduction process may appear to place an extrburden upon the teacher, in practice the extrtime required to reproduce the material is morthan compensated for by the positive results, bomathematically and behaviorally, that appear

8

the classroom.

if,'"

Finally, early in 1970, SMSG convened anotherconference to discuss mathematics programs ininner -city schools. During this conference, mostof the more promising inner-city school mathe-matics programs were reviewed and discussed, anda number of suggestions were presented for waysof improving these programs. A report of,this con- ,

ference is available'.

Obviously, SMSG has merely scratched the sur-ace of the problem of providing suitable mathe-atics programs for disadvantaged and low

chieving students. We hope, however, that thectivities will point the way to more numerousnd more powerful efforts in the future.

REPORTSThis series consists of reports, too long to be in-

cluded in SMSG Newsletters, on various SMSGprojects. Single copies may be obtained by a post-card request to SMSG, Cedar Hall, Stanford Uni-versity, Stanford, California 94305.

2. The Special Curriculum Project1. The Special Curriculum Project: 1965-665. The Slow Learner Project: The Secondary

School "Slow Learner" in Mathematics6. Preliminary Report on an Experiment with

Junior High School Very Low Achievers inMathematics ,

7.. Final Report on an Experiment with JuniorHigh School Very Low Achievers in Mathe-matics

ORDER INFORMATION

1. Accredited schools will be given an educational discount of30%. Shipping charges will be billed to the purchaser.

2. Other institutions or individuals interested in mathematicseducation will be given an educational discount of 30% onorders totaling $10 or more. Shipping charges will be billedto the purchaser.

3. Orders totaling less than 910, and not from accredited schools,will be billed at list price, but shipping charges by book postwill be paid by A. C. Vroman.

4. Orders from individuals should be accompanied by remittance,including 5% sales tax on orders originating in California.

5. Orders from overseas accounts which have not establishedcredit should be accompanied by remittance.

6. We regret that we are unable to supply free desk and exam-ination copies.

7. Returns may not be made without prior permission.

8. We believe that it is vital to the success of the individualstudent as well as to the planned curriculum of the entireclass, that Teacher's Commentaries not be sent by us to stu-dents. We therefore request that teachers ordering Commen-taries for examination please do so on school stationery or insome other way to indicate professional status.

9. As it takes longer to process an order during the rush perkalin July, August, and September, we urgently suggest that-youplace your order well in advance of your needs. Orders will beshipped on a first come, first served basis.

10. All correspondence concerning orders should be addressed to

A. C. Vroman, Inc.20851. Foothill Blvd.Pasadena, California 91109

10

IN MATHEMATICS

Inservice Course for PrimaSchool Teachers ISM13) $2.50

ma REPO TSConferenceon Future Responsibilities

for School Mathematics (CR-5) .... $ .50Mathematics Education for

Below Average Achievers. (CR-61 .... $1.00Report of a Conference on. Mathe-matics Education in the InnerCity Schools (CR-8) $ .75

OH OARY SCHOOL. MATHEMATICSIii EDITIONS

Chapters 1, 2 and 3, Student's Text $2.00Chapters-4 and-5,-Student's-Text :-.1.-- $2.00Chapters 6 and 7, Student's Text..;. $2.00Chapters 8 and 9, Student's Text .... $2.00Teacher's Commentary for Chapters1-9

$2.00

THIAlleM1311 THE ELEMENTARY. SCHOOL

_._.. Book K, Teacher's Commentary $1.00L--.: Book 1, Student's Text, Parts I and II $1.50--; Book 1, Teacher's Commentary,Parts I and 11 .. $3.00

Deveiopin Mathematics Readinessin Pre-School Programs $ .75

A. C. Woman, Inc.4485 E. Foothill Blvd.Pasadena, California 91109

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41

tHoOL-ATHEMATICSTIODY GROUP

ewsletter No. 36bruary 1972

INAL REPORT ON A NEW

URRICULUM PROJECT

1

FINAL REPORT ON ANEW CURRICULUM PROJECT

Is of the Projectrom 1958 to 1966, SMSG devoted the majort of its efforts to the preparation of sampletbooks. Of course many other activities were

undertaken preparation of materials forthem and, for the more able students, evalua-s of various sorts, etc. but all these togetheresent less time and manpower than the text-k effort. With the revision of a computer text

ring the summer of 1966, a complete set of ,

ts was available covering the-entire range fromdergarten to the end of high school.n 1966, upon the direction of its Advisory

d, the School Mathematics Study Group ini-ted the development of a new flexible, sequen-I--program -m mathematics for-grades- seven--

ugh twelve. The primary emphasis was de-ed to developing a new junior high curriculum,ondary School Mathematics (SSM), which capa-

students can complete within three years,le less capable students may profitably spendr or more years completing it. New ideas andcepts are refined and generalized in a spiral

nner throughout this sequence.istinguishing characteristics of SSM are:this curriculum is devoted solely to those

- mathematic-al- concepts- which-the-SMSG-planning and writing groups believe all citizensshould know in order to function effectivelyin our society;the usual grade placement of mathematicaltopics is ignored; instead, topics from arith-metic, algebra, and geometry are introducedin a logical sequence and in such a way as toprovide mutual support;

icertan topics new to the junior high schoolcurriculum are included; in particular, func-tions, coordinate geometry, rigid motions,

:computer mathematics, probability, and sta-- tistics;

strong attempt is made to make clear to the, students the relevance of mathematics to prob-lems of the real world.nit- the-aim- of-SS M-was- to-include- those-air

which all citizens should know, some topicsare commonly taught in the first year of

bra are purposely postponed. These topics,formal deductive development of geometry,

di topics from Intermediate Mathematics areated in a follow-up course, Secondary School _

:faced Mathematics (SSAM), which is de-

signed for students who elect to take additionmathematics after completing the junior hisequence. SSAM prepares the student for a van(of courses: Analytic Geometry; Elementary.Fmtions and Calculus; Algorithms, Computaticand Mathematics; Matrix Algebra; and Probatity and Statistical Inference.

Development of the CurriculumThe development of the. new sequence beg

with letters sent to Science and to The Mathemics Teacher early in 1966 which announced tlnew project and urged all those interested in whmathematics is taught in our schools to su!::rsuggestions and comments. Several hundrthoughtful letters were received and studiedthose involved in the project.

During the summers of 1966 and 1967, a groOf mathematicians and secondary school teachwrote detailed outlines of the .proposed progrfor grades seven through nine and discussedternative proposals for curricula for grades tthrough twelve. These outlines were based on trecommendations of a panel of university matmaticians, . applied mathematicians, and pracing secondary school teachers, which had metthe spring of 1966 in New Orleans.

A small group of writers prepared experimenversions of fourteen chapters during the 1966academic year based on the detailed outlinParts of several chapters were rewritten after tmaterials had been tried in seventh grade drooms. During the 1967-68 school year theseperimental chapters were tried out in sixteenentli grade classes. Most of the classes finishedtween ten and twelve of the proposed chapters.general, the reactions of the students, the paresand the teachers were quite favorable. Teacand student evaluations were .systematicallylected and these evaluations served as a basisthe revision of nine of the chapters duringsummer of 1968.

During the academic year 1967-68 anotgroup of writers prepared experimental versiof fourteen more chapters-These chapters wtried out by the same group of students and teaers iirthe eighth grade during the 1968-6 s-racyear.. The same group of teachers and anotgroup of seventh grade students tried out pliminary versions of the first twelve chapters ding the 196-69 school year.

During the summers of 1969, 1970, and 19other writing teams met to continue makingvisions based on teacher and student evaluatio

4

the versions used during the previous academic. Classroom tryouts continued during the aca-lc years of 1969-70 and 1970-71. New teach-were involved in the project as the students

ved from junior high to high school. By theof 1970, the first eight chapters were pub-

ed in their final form. The remaining twentyters were published in their final form in

he following people participated in planningwriting sessions for SSM and SSAM.

ela Ames, University of ChicagoG. Beg le, School Mathematics Study Group

S. Bell, University of Chicagovid Blakeslee, San Francisco State Collegessell N. Bradt, University of KansasCreighton Buck, University of Wisconsinn M. Calloway, Kalamazoo College ----Warn G. Chinn, San Francisco Unified School'strict

ruin H. Colvin, Boeing Scientific Researchboratoriese Corcoran, California High School, Whittierand E. David, Jr., Bell Telephoneboratoriesrd A. Dean, California Institute of

ethnologyHam S. DeVenney, School Mathematics Studyroupes M. Dobbie, Arthur D. Little, Inc.Eugene Ferguson, Newton High School,ewtonville, Mass.

her Fleming, Hamline Uaiversity, St. Paulnard Gillman, University of Rochesterhard A. Good, University of Marylandbert J. Greenberg, University of Denverh T. Herriot, Gunn High School, Palo Alto

n J. Hoffman, International Businessachines

non R. Hood, Portland Public Schoolsley B. Jackson, University of Maryland

ore S. John, Laboratory Schools, University ofcago

1 S. Jorgensen, Carleton College, Northfield,inn.iic1:Juncina7The-Rand-Gorporatio:a

dred Keiffer, Cincinnati Board of EducationHam G. Lister, University of New York, Stonyrookes C. McCaig, Cupertino School. District,upertino, Calif.

Mary Ferrer McFarland, Hayward Stateliege, Calif.

Thomas M. Mikula, Phillips Academy, HanovN.H.

H. Stewart Moredock, Sacramento State CollLowell J. Paige, University of California at

AngelesMax Peters, Wingate High School, Brooklyn,B. J. Pettis, University of North CarolinaHenry 0. Pollak, Bell Telephone LaboratoriWalter Prenowitz, Brooklyn CollegeRichard S. Presser, Michigan City Schools,

IndianaPersis 0. Redgrave, Norwich Free Academy,

Norwich, Conn.George Richardson, School Mathematics Stud

GroupDonald E. Richmond, Williams CollegeEric M. Rogers, Physical Sciences Study _

CommitteeSidney Sharron, Los Angeles City School DistrRichard W Steffen, Fremont High School,

Sunnyvale, Calif.Warren Stenberg, University of MinnesotaJane G. Stenzel, Cambrian Elementary Schoo

District, San Jose, Calif.Henry W. Syer, Kent School, Kent, Conn.Sally H. Thomas, School Mathematics Study

GroupM. L. Tomber, Michigan State University

_William_Wernick, City College of New_York____Hassler Whitney, Institute for Advanced Stu

PrincetonRobert P. Wilens, Bret Harte Junior High Sch.

San Jose, Calif.Gail S. Young, Tulane UniversityMartha Zelinka, Weston High School, Westo

Mass.Description of the Curriculum

The junior high sequence, SSM, has the lolling features.

1. An attempt is made to fuse arithmetic, abra, and geometry in such a way that eachhelps in supporting the development of the otwhenever possible.

2. Geometry is presented in 'a concrete, intive, descriptive way. When a new concepintroduced,- the- focus -is on- its-essential-featuIts more complex aspects and its relation to otconcepts are treated later. The treatment ofometry fosters understanding of the concreteand the intuitive significance of geometricbefore they are studied on a rather formal dtive basis.

3. One, two, and three dimensions aresidered in many geometrical questions w

6

plicable. Coordinate geometry appears as ap-ropriate in helping describe sets of points alge-raically. Solution sets of algebraic equations areterpreted or described geometrically.4. The process of model building in appliedathematics is the construction of a mathematicalodel that will help one better understand a situ-ion that occurs in a complex environment, sinceie analysis of such a mathematical situationten enables one to learn more about the origi-1 physical situation. This process is developedappropriate places in the sequence.5. Some relaxation in the stress on structure isticeable, but structure is still definitely one ofe unifying themes.6. Short deductive sequences appear using prop-ties discovered in the intuitive development of

h-- algebra and geometry. Various methods ofoof are illustrated.7. Appropriate materials are included so thate student has some appreciation for the mathe-atical activities associated with computer used the role of computers in our technologicalety. Mathematical algorithms expressed in

w chart notation oiler an introduction to com-ter activity and require the student to have

fficient understanding of the mathematics foractical exploitation.-8.-The_ concept of function is considered earlyd used in many diiii-ent-tylits-of-mathematical

tent whenever appropriate.9. The development of geometry culminatesth a chapter on rigid motions and vectors.10. Three chapters introduce probability andtistics and lead up to some examples of hy-thesis testing.As with all SMSG texts, each chapter is ac-.mpanied by a Teacher's Commentary. In thisuence, each commentary describes why the

apter was included and how it fits with theher chapters, states the purposes and objectivesthe chapter, gives a suggested time schedule,

ovides additional background for each sectiond answers to all exercises, and concludes withggested test items and answers.T-ile-SSM-sequence is published in fourteenits with two chapters to a TiPur.-Tlie-thapter

les are:

1. Structuring Space2. Functions3. informal Algorithms and Flow Charts4, Problem Formulation5. Number Theory

7

6. The Integers7. The Rational Numbers8. Congruence9. Equations and Inequalities

10. Decimal Representation for RationalNumbers

11. Probability12. Measurement13. Pernendiculars and Parallels (I)14. Simi, larity15. The Real Number System16. Area, Volume, and Computation17. Perpendiculars and Parallels (II)18. Coordinate Geometry19. Problem Solving20. Solution Sets of Mathematical Sentences21. Rigid Motions and Vectors22. Computers and Programs23. Quadratic Functions24. Statistics25. Systems of Sentences in Two Variables26. Exponents and Logarithms27. Logic28. Applications of Probability and StatisticsThe SSAM sequence consists of eight chapte

bound in five units, followed by IntermediatMathematics, Part II. The chapter titles are:

1. Organizing Geometric Knowledge2. Concepts and Skills in Alegbra3. Formal Geometry4. Equations, Inequalities, and Radicals5. Circles and Spheres6.. The Complex Number System7. Equations of the First and Second Degree

Two Variables8. Systems of Equations

9. Logarithms and Exponents10. Introduction to Trigonometry11. The System of Vectors12. Polar Form of Complex Numbers13. Sequences and Series

14. Permutations, Combinations, and theBinomial Theorem

Optional chaptersSuggested Sequences and Grade Placements

A primary goal of this curriculum project is-provide-a-flexible-secondary-school-mathemat

program. The typical plan is for average aabove average mathematics students to begin SSin seventh grade. Students will vary in the lenof time they take to complete SSM. Those sdents who plan to take further mathematicsthen take SSAM which provides a transition tvariety of more advanced courses. Teachers

8

ect from Intermediate Mathematics, Part 11,ose chapters which are needed for the remain-g mathematics courses their students will take.lose classes which complete the SSM and SSAM

quence by the middle of the junior year have ariety of possibilities for the remaining threemesters.Possible sequences of SMSG courses for teachersselect from depending on the needs and inter-

ts of their students are:Elementary Functions for one semester fol-lowed by Calculus for one year which wouldprepare students to take the BC AdvancedPlacement Exam in calculus;For those who wish to take more mathematicsbut not calculus, one semester each of Intro-duction to Matrix Algebra; Algorithms, Com-putation, and Mathematics; and a course inProbability and Statistical Inference using oneof the commercially available texts.Calculus of Elementary Functions for one year,which would prepare students to take the ABAdvanced Placement Exam in calculus, andone of the semester courses in option 2.Classes that complete SSM and SSAM at theend of their junior year may choose:Calculus of Elementary Functions for their lastyear, ortwo of the three semester courses listed in op-

Slower mathematics students may study theM sequence for a longer period. Since SSM issigned to contain the mathematics which allizens should know, students who begin the se-ence should continue until they have com-

eted it. It is of interest to note that in at leaste high school, the lowest achievers in mathe-tics are beginning SSM as ninth graders and

11 spend four years studying it.suits of the Classroom Try-outIn 1967-68, 527 seventh grade average or aboveerage mathematics students from middle classjo-economic backgrounds began studying SSM.

ey were in fifteen classes taught by fifteen dif-ent teachers in five schools in or near San Jose.these students, 85% remained in the program

rough-the-eighth grade, 58% through the ninthde, and 37% through the tenth grade. In

70-71, there were eight tenth grade classes inn high schools. These classes completed the

M sequence in the fall and finished about halfthe SSAM sequence. Some schools have nowased those students who are continuing in

9

mathematics into their regular eleventh or twelftgrade mathematics classes, while others are Cotinning with SMSG courses.,

In 1968-69, another similar group of 448 seenth graders taught by the same teachers begastudying SSM. 86% of this group remained in tprogram through the eighth grade, but only 19were in it at the end of the ninth grade. Mostthe students who dropped out after two yeawere entering high schools which had dropthe program. This decision was made becauthere were too few SSM students attending thehigh schools to make up full classes.

The try-out students were tested in the falltheir seventh grade year and in the spring of eayear they remained in the program. Many of tltests that were administered to SSM students halso been administered to students in the five yeNational Longitudinal Study of MathematicAbilities (NLSMA) from 1962 to 1967. Statisticcomparisons were made between the SSM studeand a sample of NLSMA students who had taka year each of algebra and geometry by the endtheir sophomore year in high school. In the fallseventh grade, the SSM students scored signicandy higher than the NLSMA students on tintelligence scales that were administered,understanding of and computation with wh

numbers,- and-on_ understanding_of_geomemrelationships. The NLSMA students scored snificantly higher on computation with fractioand decimals.

At the end of the ninth grade, the NLSstudents scored significantly higher than the SSstudents on knowledge and understanding of albraic expressions and equations. This is toexpected since most of the work on algebrmanipulation occurs in SSAM (tenth grade fthese students). By the end of tenth grade, SSstudents scored higher on algebraic expressiothan NLSMA students who had taken this testninth grade but lower than those NLSMAsdents who took it again at the end of elevengrade after studying a second year of algebSimilarly, when the SSM students were testedthe-end-of-ninth-grade-on-geometry,-they-Scorlower than NLSMA students given this test at tend of tenth grade. But when the SSM studewere given this test again at the end of tengrade, they scored higher than the tenth graNLSMA students had. When both groups wtested at the end of the same grade, SSM studescored higher than NLSMA students on algebr

10

equalities (ninth grade) and on measuremententh grade). The two groups had the samecan on a two-item scale on structure of proofsministered at the end of tenth grade.The SSM students were also given some slat-ally constructed tests designed to evaluate meth-s of developing topics that were used in certain

iapters to see if these methods were effective.esults of these tests were given to the writers to

while revising the corresponding chapters.The try-out teachers attended biweekly semi -ts by SMSG staff members. These

minars included discussion of the content of theiapters in the sequence and methods the class-

m coordinators had found useful in teachinge pilot class. Time was also devoted to discuss-

how the chapters currently being taught wereng. Notes were made of the teachers' conv-

ents. The teachers also turned in written evalua:ns on each chapter. This information was usedthe writers when the sequence was revised the

'lowing summer.At the end of the last seminar, the tenth gradey-out teachers were asked to describe their stu-nts. They were especially pleased with the per-rmance of their students in reasoning, under-nding verbal problems and setting up equa-ns, understanding proofs, and in recalling and

ing_previously developed concepts.The coordinators oftheClarcieliftlyciiirpliase---this project were William S. DeVenney and

mes C. McCaig. Their work included teachingpilot class which studied the SSM chapters be-.re the try-out classes did, in order to warn the

out teachers of possible pitfalls. This class at-ded Ortega Junior High School in Cupertinoion School District and Fremont High SchoolFremont Union High School District.

The following .Northern California school dis-cts, schools, and teachers participated in thessroom try-out and data collection of SSM andAM.

Campbell Union High School DistrictBranham High School, San Jose

Robert B. CanihanLeigh-High-School i-Sanclose

Edwin D. StoneCupertino Union School District

Joaquin Miller Junior High School, San JoseJames D. HughesJames C. McCaigKenneth B. McNeal

11

Fremont Unified School DistrictHopkins Junior High School, Fremont

Gordon W. BoikeJoan M. CowgillDelbert V. Sullivan

Mission San Jose High School, FremontGeorge E. CrozierGeorge A. Tomlinson

Fremont Union High School DistrictLynbrook High School, San Jose

Stanley R. BooneOtis T. HallidayRobert L. Stone

San Jose Unified School DistrictBret Harte Junior High School, San Jose

Howard G. Ester lineJane R. PflughauptRobert P Wilens

Leland High School, San JoseArt DeVault

Santa Clara Unified School DistrictEmil R. Buchser High School, Santa Clara

Ronald M. HagelinSanta Clara High School, Santa Clara

Kent G. MarshallWilliam A. Wilson Intermediate School,

Santa ClaraRalph R. BesslerRobert.J.Culbertson_Gary It Leitz

Union School DistrictDartmouth School, San Jose

James C. HaugaardCharles S. HogdonClarence Kludt

In addition, several schools in the Los AngelCity School Districts used SSM for two years witaccelerated students under the direction of ClyCorcoran. One school in San Diego County eperimented with using SSM in various settingscluding a mathematics laboratory and small grouinstruction. This attempt was directed by ThomP. Davies. Feedback from these experiences .wcollected and distribi tied to members of the wriing team.

ImplementationThe implementation of the SSM program

volves a commitment of at least three years,ginning with Chapter 1 and starting generally.the 7th grade. This is due to the fact that topiusually taught in isolation are intermingled acarefully sequenced. Therefore, it is imperati

12

ithe teachers involved in the program becomeoughly familiar, ith the entire program, espe-lythe portion that precedes the part they areently teaching.

e SSAM program is designed to provide asition for the student from the SSM program

a more conventional mathematics program inlatter part of high school. SSAM has beenpied so that it may be flexibly implementedpiling: upon the particular school, situation.

aChers may select from this program portionsiCh will best facilitate a transition to whatever

Oh their school offers to students who haveceisfully completed SSM.t is highly recommended that some in-service'fling program be provided for teachers who

be- teaching SSM. The Teacher's Commen-es that are provided for each of the chaptersrather extensive and should be helpful refer-ei'in such a program..They contain notes re-ding the role of certain, class discussion exer-

the .use of the spiral approach for certaincepts and the nature of any ipeCial develop-tylised.

orinal courses for in-service teachers beingned to teach SSM could concentrate on theowing four areas-which correspond to the cen-themes of the SSM program. (The correspond-

-SSM .chapters are given in parentheses under_theme.)

The Number Systems of Arithmetic and Alge-bra.Stress would be placed On the structure. ofthe::systems'of non-negative integers; the irate=ger rationals, and the reals as done inSSM.:Vincent Haag's Structure.of Elementary4Igebra, Studies in Mathematics; Whittle 8, .

SMSG, could be used as a' reference (SSM:Chapters 5-7, 10, and 15). Work on algebra'should be included as needed, with polynomi-alrn one variable over a given system possi=bly

,iserving as a final topic (SSM Chapters 9,

20;23, 25, and 26).etrySSM program contains a great deal of

nietry, including some coorclinate:geome-trd motions, right, triankle trigonometry,nd yectora. The SSM geometry chapters could

be analyzed carefully using the SMSG 'text,GeOrnetry, as a referenc.and some attentiongiven to the ride of logical inference, un-defined concepts, definitions, postulates, andtheorems in geometry. Finally, the SMSG

13

.:;::,

Geometry With Coordinates, could be usedbackground for some coordinate geomet(SSM Chapters 1, 8, 12-14, 16-1.8, 21, and 2

C. Probability and StatisticsThis subject will.be new to most teachers. Tthree SSM chapters on this theme couldstudied carefully using an introductory stattics and probability text as a reference (SSChapters '11, 24, and 28).

D. Problem Solving and Computer MathematiA great deal of attention is given to problsolving and model building (SSM Chapters4, and 19). Likewise, there is a substantial ephasis given to the use of flow charts and coputers (SSM Chapters .3 and 22). Theretothese chapters should be analyzed carefullying the references mentioned in the TeacheCommentaries as a source for supplementawork.

Teachers who have had experience teachicontemporary Plane Geometry and Algebra ITrigonometry courses should need no in-serytraining for the SSAM program. The plannitime for implementing this program shoulddevoted to a familiarization with the contentthe chapters and to a careful selection of thchapters most appropriate for a transition 'to tmathematics courses offered in the final semest,of high school.

Description of a Low Achiever Version of SSMIn the summers of 1969, 1970, and 1971, oth

writers prepared a Special Edition of SecondSchool Mathematics for seventh and eighth gralow achievers. This edition was based on findiOf low achiever studies by Sarah T. Herriot aWilliam S. DeVenney in SMSG Reports, Nos.6, and 7. The chapters, in this program introdthe same general content areas as the first 1years' work in the version for average and abiaverage mathematics students, but the nutterhas been rewritten to appeal to students whymathematics achievement in elementary schiwas very low. The quantity of reading and treading level have been reduced to a minimuThe student is provided with addition and muplication tables and with flow charts for comptitional algorithms. Thus, he is relieved of as mcomputation as is possible. and is free to conctrate on mathematical conceptS and relationshiThis program consists of the following chaptwhich can be covered in approximately two ye

1. Flow Charts2. Structuring Space

14

3. Functions4. Number Theory5. The Integers6. Rational Numbers7. Probability8.` Equations9. Congruence

10. Decimals11. Parallelism12. Similarity13. More About Rational Numbers

;:14. Perpendiculars15. Measurement16. Real Numbers17. Solving Equations and Inequalities18, Coordinate Geometry

Teacher's Commentary accompanies each set ofne chapters.The material is presented to the student in therni of work sheets; however, it is not, in the or-nary sense, a work book. Topics in each chapter

developed sequentially lesson by lesson. Teach-4ed class discussion exercises are carefully pro-awned to lead the student to successful exile-noes in the exercise sets. Each student shouldprovided with a binder in which to place his

ineeted lessons. This relieves the student ofping track of a textbook and allows him to

ncentrate on what' he has accomplished, rathern on what remains to be completed. Each

apter concludes with (1) a practice test whichntially tells the student what he is expectedknow; A2) a chapter test which parallels theaioe test; and (3) review exercises to promote

tendon of material covered in previous chapters.As the cost of producing individual worksheets

mercially proved to be prohibitive, this ma-al comes to the teacher in bound volumes.

di chapter consists of a number of lessons ofe or more pages. Each page is perforated forsy removal. These pages can be reproduced inantities sufficient for the class by means of spiritstets. The cost of reproducing these materialsthe school is offset by the need to purchase only

set of student texts.

oom 'Pry-out of Low Achiever ProgramThe preliminary version of Secondary Schoolathematics, Special Edition, was tried out in69-70 and 1970-71 in seventh and eighth gradeImes of low achieving mathematics students lo-ted in San Francisco, Oakland, San Jose, andorthern suburbs of San Jose. The classes car-ted of students who were about two- years be-

low grade level in mathematics. Sixteen seventgrade classes with fourteen different teachers ieleven schools initially participated in the, studEleven of these classes in ten schools continuein the study when their students were eightgraders. Several classes consisted of disadvantagemixed urban minority (including Black) studenand two classes were primarily Mexican- AmericaMost eighth grade classes completed sixteen of thchapters in the sequence.

During the two years of the study, the teachesattended biweekly seminars which covered tcontent of the chapters (some of which was neto most of the teachers) and the philosophy of tprogram. Problems which arose during the try -otwere discussed. Teacher evaluations served asbasis for the final revision of these chapters tfollowing summer. The reactions of the teaters and students were extremely favorable. Tteachers commented particularly on how the stdents' attitudes toward mathematics and theself-concepts had improved during the two yeathey were in this program. Many more studenthan usual elected to enroll for more or highlevel .mathematics as ninth graders than wasquired., Students who complete the Special Elion chapters are well prepared to study a coursuch as SMSG's Introduction to Algebra overtwo-year period.

16

The following school distriCts, schools, andchefs participated in the classroom try -out of

is project. .

Cupertino Union School DistrictJoaquin Miller Junior High School,

San JoseAnn L. Bixby

Ortega Junior High School, SunnyvaleL. Hart Andelin

TWereland School DistrictRogers Junior High School, San Jose

David J. HallstromOakland Public Schools

Frick Junior High SchoolKatherine Maze

Hamilton Junior High SchoolJimmie Neblett

San Francisco Unified School DistrictDenman Junior High School

Marie J. FaraoneEverett Junior High School

Frances LeechStephen Malayter

Benjamin Franklin Junior High SchoolRobert Kane

A. P. Giannini Junior High SchoolJoseph L. Glikman

Herbert Hoover Junior High SchoolKatherine BuckJo Ann McNallyWilliam Ota

an Jose Unified School DistrictPeter.Burnett Junior High School

Preston Luke

17

ORDER INFORMATION

1. Accredited schools will be given an educational discount30%. Shipping charges will be billed to the purchaser.

2. Other institutions or individuals interested in mathematieducation will be given an educational discount of 30%orders totaling $10 or more. Shipping charges will be billto the purchaser.

3. Orders totaling less than $10, and not from accredited schoowill be billed at list price, but shipping charges by bookwill be paid by A. C. Vroman.

4. Orders from individuals should be accompanied by remittaincluding 5% sales tax on orders originating in California.

5. Orders from overseas accounts which have not establishcredit should be accompanied by remittance.

6. We regret that we are unable to supply free desk and eination copies.

7. Returns may Mt be made without prior permission.

8. We believe that it IR vital to the success of the indivistudent as well as to the planned curriculum of the eatclass, that Teacher's Commentaries not be sent by us todents. We therefore request that teachers ordering Comtaries for examination please do so on school stationery osome other way to indicate professional status.

9. As it takes longer to process an order during the rush pein July, August, and September, we urgently suggest thatplace your order well in advance of your needs. Orders wilsnipped on a first come, first served basis.

10. All correspondence concerning orders should be addressel

A. C. Vroman,2085 E. Foothill Blvd.Pasadena, California 911

18

DER FORM

Its

SCI1001. MATNEUTICSChapters 1 and 2, Student's Ted $ .75Teacher's Conimentary ..... $1.00Chapters 3 and 4, Student's $ .75Teacher's Comlnentary $1.00Chapters 5 and 6, Student's Text .... $ .75Teacher's Commentary ..:_........ $1.00Chapters 7 and 8, Student's Text $ .75Teacher's Commentary $1.00Chapters 9 and 10, Student's Text.... $ .75Teacher's Conintary $1.00Chapters 11 and 1, Student's Text $ .75Teachees Commentary $1.00Chapters 13 and 14, Student's Text $ .75Teacher's Commentary $1.00Chapters 15 and 16, Student's Ted $ .75Teacher's Commentary $1.00Chapters 17 and 18, Student's Text $ .75Teacher's Commentary $1.00Chapters 19 and 20, Student's Text $ 75Teacher's Commentary $1..00Chapters 21 and 22, Student's Text $ .75Teacher's Commentary $1 00Chapters 23 and 21, Student's Text $ .75Teacher's Comment $1.00Chapters 25 and 26, Student's Text $ .75Teacher's $1.00Chapters 2Ciem anlicilerntudent's Text $ .75Teacher's Commentary $1.00

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0Teacher's C ntaommery $1.0

SUNK AIRTNEIMINSOITtONt_Chapters 1, 2 and 3, Student's Text $2.00

Chapters 4 and 5, Student's Text .... $2.00Childers 6 end 7, Student's Text .... $2.00Chapters and 9, Student's Text $2.00Teachers Commentary for Chapters

1-9 $2.00*Chapters 10 and 11, Student's Text $2.00Chapters 12, 13, and 14,

Student's Text $2.00Chapters 15 and 16, Student's Text $2.00Chapters 17 and 18, Student's Text $2.00

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TE MADINAT=Student's Text, Part II $1.50Teacher's Commentary, Pert II ..... $1.50

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41

-SCHOOLMATHEMATICSSTUDY GROUP

Newsletter No. 38

August 1972

Section IAn SMSG Statement on Objectives

in Mathematics Education

Section IIMinimum Goals for Mathematics

Education

rd

SECT ION IAN SMSG STATEMENT ON OBJECTIVES IN

MATHEMATICS EDUCATION

IntroductionCareful attention to objectives of mathematics

education has, very recently and very suddenly,become much more important than was the casein the past. The reason for this is that dissatisfac-tion with the results of our educational systemhas become widespread and has led to detripndsfor much more accountability on 'the part ofschool systems and teachers than in the past. Ac-countability, however, can only be meaningfulwhen actual achievement is compared with hoped-for achievement, and to specify the achievementswe hope for is to state the objectives of education.

One result of this concern has been pressureon schools and on teachers to state their educa-tional objectives in enough detail so that the com-parison can indeed be made. It is a fact that suchan exercise is unfamiliar to most teachers and noteasy to carry out, and much of what has beendone recently is seriously deficient. Rather thandwell on the details of these deficiencies, we takethis opportunity to indicate some impoitant cri-teria that should be kept in mind when examiningstatements of educational objectives.

It must be admitted that SMSG has not in-dulged in the statement of educational objectivesto any great extent, particularly in its curriculumdevelopment activities. When SMSG started workin 1958, a deliberate decision was made not tostart by preparing detailed outlines of, and ob-jectives for, the courses it was to prepare. Therather general outlines provided by the Commis-sion on Mathematics of the CEEB, and the beliefthat understanding of mathematics was just asimportant as computational skill, were felt to beenough to point the way. And indeed, in each ofthe SMSG curriculum development projects, thesame philosophy and a rather broad outline wereall that were given to the writing teams. In viewof the success of each of these projects, such quid -.ante seems to have been enough.

On the other hand, when, in the early sixties,it became apparent that a careful, intensive eval-uation of the various SMSG curricula was needed,the matter of the objectives of the various mathe-matics curricula then available became para. 4*mount.

The particular answer developed for the Na-tional Longitudinal Study of Mathematical Abili-

3

ties was to specify objectives by means of sets ofspecific test items. These have all been publishedin Wilson (1968). Background information con-cerning the rationale and development of thesetests can be found in Romberg and Wilson (1969).

In any case,.we hope that the long preoccupa-tion with educational objectives in the past notonly of the NLSMA staff and consultants but alsothe SMSG Advisory Board may have contributedusefully to the following discussion.

The original review and analysis of the litera-ture was carried out in a seminar directed byDr. E. G. Begle and attended by ProfessorsNicholas Branca and Decker Walker. The follow-ing graduate students of the Stanford UniversitySchool of Education attended: Len Berk, GeorgeEshilvani, William Geeslin, Joseph Green, RogerJarvis, Helen McCullough, Bertrand Morin, Bar-barn Pence, Stanley Puryear, Sally Thomas, Nor-man Webb and Robert Wise.

Variation in Statements of Educational Or jectivesGoals and objectives of mathematics education

have been put forth by many individuals, ad hoccommittees, and national organizations. We willnot attempt to review, even sketchily, these manylists of objectives. Very useful reviews are to befound in the dissertations of Harding (1968) andOakes (1965). The NCTM (1970) publication onthe history of mathematics education in the U. S.is another useful reference.

A hasty look at even a few of these statementsis enough to reveal that they have been phrased in

a wide variety of styles. One result of this is thatit is usually difficult to compare one list of objec-tives with another. A second result is that manystatements of objectives are actually quite mean-ingless and hence useless.

However, a more careful look at, especially, thelists of objectives presented by the major com-mittees over the, years indicates that statements ofobjectives vary over a limited number of dimen-sions. A close look at each of these dimensionswill be helpful.

1. ContentThe first and most obvious dimension along

which educational objectives can vary is that ofthe mathematical topic to which the objective

A preliminary draft of this document was reviewed bythe SMSG Advisory Board at its final meeting in April,1972. The present version incorporates many suggestionsand comments made by Board members.

4

refers. The topic might be arithmetic, or geo-metric, or a part of analysis, or logic, etc. Hereare some examples:

Mastery of the four fundamental operationswith whole numbers and fractions, written indecimal notation and in the common notationused for fractions.

Commission on Mathematics (1959), p. 19.Given a numeral in any base, two-twelve, the

students will find the corresponding base tennumeral.

Instructional Objectives Exchange (1968), p.14, Grades 7-9.

The ability to recognize, to name, and tosketch such common geometric figures as therectangle, the square, the circle, the triangle, therectangular solid, the sphere, the cylinder, andthe cube.

Joint Commission of the MAA and NCTM(1940), p. 54.

The child should learn to estimate by the eyeand to measure with some degree of accuracyof lengths of lines ...

NEA (1894), p. 110.Boys must learn to draw general figures; tri-

angles must not be drawn Isosceles, quadrilat-erals should not be drawn as parallelograms.

Mathematical Association (1959), p. 68.-Introduction in grade 11 of fundamental trig-

onometrycentered on coordinates, vectors, andcomplex numbers.

Commission on Mathematics (1959), p.The student should understand and be able

to correctly use the terms:simple statement, compound statement, nega-tion, conjunction, disjunction, conditional,biconditional, logically equivalent, converse,contrapositive, direct proof, proof by contra-

' diction, quantifier.SMSG (1971), Chapter 27, teacher's commen-

..'.taty, p. 2.Appreciation of mathematical structure

("patterns")for example, properties of nat-ural, rational, real; and complex numbers.

Commission on Mathematics (1959), p.

Although some of the variations may be attrib-uted to grade level, .nonetheless the above illus-trations are enough to remind us that a wide va-riety of mathematical topics have been thought tobe desirable for the school curriculum. Even so,

5

they are not sufficient to demonstrate the verylarge number of topics which actually have beenadvocated at one time or another.

In addition to specific topics, such as those il-lustrated above, much broader descriptions ofcontent are often made.

Every phase of the mathematics curriculum,as well as all classroom procedures, would bescrutinized for. the opportunities to develop agrowing appreciation.of the immense power ofmathematics, of its record as a universal servantof mankind, of its cultural significance, and ofits permanent place in the study of nature, inscience, in the practical arts, in business, en-gineering, and everyday life.

Joint Commission of MAA and NCTM(1990), pp. 68, 69.

. . . pupils will have gained an actually realinsight into the true nature of mathematics andan enthusiasm for it.

joint Commission of MAA and NCTM(1990), p. 74.

To see mathematics as a human endeavorwhich demands creative energy.

Beberman (1958), p. 38.Understanding of the nature and role of de-

ductive reasoning in algebra, as well as geom-etry.

Commission on Mathematics (1959), p. 33.

2. Cognitive levelWhen a student adds two three-digit numbers

he is working at a different cognitive level (or isusing a different cognitive process) than when heselects from a set of geometric shapes those whichare not convex. When he locates the maximumvalues, over an interval, of a particular polynomialfunction, he is working at still another level.

There are many schemes for the categorizationof cognitive levels. However, most of them haveevolved from the taxonomy defined by Bloom(1956). The categorization scheme which will be

iused in this report is essentially the one used inthe NLSMA study. This scheme also evolved fromthe Bloom Taxonomy and, as in the taxonomy,each category is more complex than the previousone. In some cases, the categories can even beconsidered as hierarchical. That is, a lower cate-gory may be a necessary prerequisite for a highercategory. The five categories which we find mostappropriate for precollege mathematics educationare Knowledge of Facts, Computation, Compre-hension, Application, and Analysis. A more pre-

6

se definition of each cognitive level category iss follows.Knowledge of FactsObjectives which requirethe recall of specifics such as terminology, sym-bols, or conventions; emphasis is on recall, notupon synthesis, generalization, or translation ofthe recalled information.ComputationObjectives which requirestraightforward manipulation of problem ele -.ments according to rules the subjects presum-ably have learned; emphasis is upon performingoperations, not upon deciding which operationsare appropriate.ComprehentionObjectives which requireeither recall of concepts and generalizations ortransformation of problem elements from one .

mode to another; emphasis is upon demonstrat-ing understanding of concepts and their rela-tionships, not upon using concepts to producea solution.ApplicationObjectives which require (1) re-call of relevant knoWledge, (2) selection of ap-propriate operations, and (3) performance ofthe operation. Objectives are of a routine na-ture requiting the subject to use concepts in aspecific context and in a way he has presumablypracticed.AnalysisObjectives which require .a non-rou-tine application of concepts. Under this headingcollie problem formuhition and mathematicalmodeling.

Categorization of objectives by cognitive levelsis complicated by other factors. For example, anobjective which is classified as computation forone age population may be classified as applica-tion or even analysis for younger populations.

An example of an objective which can be cate-gorized as Knowledge of Facts is:

Ability to know and name correctly the sphere,cube and cylinder and the most characteristicsurface forms such as circles.

International Commission (1911), p. 286.An example of the Computation category is:

To perform the fundamental operations withfractions.

Breslich (1930), p. 205;of the third category of Comprehension is: .

. . . the. student should understand that the-

'Description of the last four categories are taken fromomberg and Wilson (1969), pp. 39 -40.

solution set of .a linear inequality such as AxBy ± C > D consists of the points in a hal

SMSG (1971), Chapter 25, p. 2.An objective of the Application category is:

'He (the student) is expected, among othethings, to solve equations (singly and in systerns), ....

Beberman (1958), p. 3.Analysis, the final : Itegory, is illustrated in th

following example.He (the student) should begin to devise con

structions and demonstrations for himself.NEA (1894), p. 115.

Many objectives include a combinition of thdifferent categories. For example, consider thilast objective which involves both Computatioand Comprehension:

The development of an understanding othe deductive method as a way of thinking an'a reasonable skill in applying this method tmathematical situations.

Commission on Mathematics (1959), p. 23.It is relatively easy to construct: tests whic

measure student achievement at the cognitive levels of Facts and Computation. However, it doenot seem to be widely appreciated that it is alpossible to measure student achievement at thhigher cognitive levels. For a full discussion othis, see Chapter 3 of Romberg and Wilson (1969).Here, however, are two sample test items, suitablfor junior high school students, to illustrate eachof the last three cognitive' levels.

Comprehension1. The product rif 356 and 7 is equal to

(A) (300 x 7) + (50 x 7) + (6 X 7)(B) 356 + 7(C) (300 + 50) + (6 X 7)(D) (3X 7) + (5 X 7) + (6X 7)(E) 300 X 50 X 6 X 7

2. This drawing suggests a rational number.Choose the fraction on the right which namesthe same rational number.

(A) 1/2 () 3/4(B) 2/3 (E) None of these.(C)

Application1. A family spent an average of $46 per month

on food for a 2-month period. If food cost $39

8

one of those months, how much was spent onfood the other month?(A) $32(B) $42.50

(D)(C) $49.50 10"

$53O$

20"2. What is the area of the shaded portion in the

figure shown above?(A) 25 square inches(B) 30 square inches(C) 50 square inches(D) 100 square inches(E) 150 square inches

nalysis1. A chess club ran a weekly tournament in

which every member played every other mem-ber just once. When one more member wasadmitted; it was found necessary to play eightmore games per tournament. Now how manymembers are there in the club?(A) 20(B) 16(C) 12(D) 9(E). 8

2. In a certain classroom, the teacher notes thatwhen the attendance is 92% there are ex-actly seven empty seats, but when the attend-ance is 88% there are eight empty seats. Whatis the total number of empty seats when theattendance is 100%?(A) five(B) four(C) three(D) two(E) none

3. The Affective DimensionMany of the objectives that have been stated

for mathematics education refer to attitudes orfeelings about mathematics. Although a taxonomyof affective educational objectives has been pub-lished, Krathwohl (1964), it has received very littleattention. In particular, very little use of this tax-onomy has been made in the design of tests orquestionnaires or in the design of research studies.Consequently, this taxonomy has had very littleempirical validation.

In addition, while there has been much interestin students attitudes towards mathematics, as in-

9

dicated in the recent review by Aiken (1969), pratically nothing seems to be known about pedagogical procedures which can be used to modifthese attitudes. While we are in agt.2ement thawe would like students to end up with a favorablattitude towards mathematics, we do not knotwhat to do to improve student attitudes, Consequently, only enough examples of statements oaffective -objectives are given .below to illustratthe wide variety that can be found among them(Note that additional illustrations appeared in theprevious two sections.)

I) convey an appreciation of the use of forminas. graphs, and simple equation.

Harvard Committee (1945), p. 162.The building of confidence in one's own ana

lytical powers.Cambridge Conference (1963), p. 9.Satisfaction with thorough work and preci

sion of statements.Smith and Reeves (1927), p. 183.Enjoyment of mathematics.National Assessment of Educational Progres

(1970), p. 30.lir feel that this activity (processes in mathe-

matics) is worthy of earnest pursuit.Beberman (1958), p. 37.An understanding of matheinatics as a contin-

uing creative endeavor with aesthetic valuessimilar to those found in art and music. In par-ticular, it would be made clear that mathemat-ics is a living subject, not one that has long sincebeen embalmed in textbooks.

Commission on Mathematics (1959), p.A degree of interest in mathematics which

will encourage the pupil to continue in thestudy.

Breslich (1930), p. 207.Ti) give the student an appreciation of postu-

lational thinking.Pre-Induction Courses in Mathematics (1943),

p. 123.Appreciation of the contributions mathemat-

ics has made to the progress of civilization.Breslich (1930), p. 206.

1. F- ifiabiliIyThis dimension refers to the degree to which

it is possible to ascertain whether or not a "'macu-lar student has achieved a particular objective.Actually, this dimension can be resolved into (atleast) three separate subdimensions. Progress

ward ati objective may be difficult, or. impossible,verify because1) the statement of the objective is ambiguous

or lacks specificity;2) we do not know how to measure progress

toward the objective;3). there are technical problems which make it

difficult to measure, with precision, progresstoward the objective.

First, let us give an example of an objectivevhich presents no difficulties along any of theseubdimensions.

The ability to express in decimal and percentorm the fractions:

1/2 3/3 3A 3/8 Vs 1/2 1 i/13

Smith and Reeves (1927), p. 191.

It is clear that an objective, to be verifiable,MISt be stated sufficiently clearly and specifically

that any two qualified evaluators can agree onhe universe of test items that would Ix! valid forests tomeasure student progress toward the oh-ective.

Variation along this subdimension actually ap-ared in the very first-two illustrations in section

I, and a review of the various illustrations in thereceding sections demonstrates that the degreeIf clarity and specificity with which objectives can

ie stated is quite large.Here are some more statements of objectives

vhich are not clear .and specific.A familiarity with the basic concepts, the

processes, and the vocabulary of arithmetic.Joint Commission of MAA and NCTM

(1940), p. 54.The objectives of a course in plane geometry .

should be to acquaint the pupil with geometricfacts and their applications and to give him anappreciation of postulational thinking.

Pre-Induction Courses in Mathematics (1943),p. 123.

IVe wish pupils to develop the conscious useof a technique of thinking.

Beatley (1935), p. 331.An understanding of the general properties

of geometrical figures Id the relationshipsamong them.

Commission on Mathematics (1959), p. 11.. . . the task of "sowing novel problems and

puzzles:

11

National Assessment of Educational Progre(1970), p. 28.

As for the second subdimension, we have alreadpointed out that we are very poorly equipped tmeasure affective variables and hence to find ohow far a student has progressed toward a ptitular affective objective. But even outside thaffective domain we have measurement problemLook back at the last three illustrative objectivin section 1. How do we find out how far a stdent has progressed toward any one of these? Siilarly, examine for verifiability the last illustrativobjective in section 2.

As to the last subdimension under this heading, refer again to the first illustrative example ithis section. Ten fractions are listed and the ojective is to be able to express each of them idecimal and in percent form, a total of twentseparate tasks. It is easy to see how to find ouhow far a student has progressed toward this o'jective: give him these twenty tasks, and obseryhow many he can do correctly.

But now let us conskier this objective:Learning column addition to six one-digi

addends and four three- and four-digit addendsFlournoy (1964), p. 10.

There are more than six trillion addition prolems which fall into the category specified in thiobjective. Obviously, we do not ask a student twork each of these problems. Instead, we eha random sample of such problems to constituta test. A student's score on such a test gives us aestimate of his achievement with respect to thobjective. The statistical theory of such estimate'is well known.

Next, consider each of these two objectives:The scholar should be thoroughly trained in

performing correctly and rapidly the four fun-damental operations with integers, vulgar 'frac-,,,,tions and decimals.

NEA (1894), p. 105.Ability to solve linear equations containini

common or decimal fractions.Smith and Reeves (1927), p. 202.

In either case, the number of problems which astudent is expected to be able to solve is infinite.There is no unique procedure for drawing a finiterandom sample from an infinite universe, andtherefore there is no well defined way of estimat-ing a student's achievement with respect to eitherof these objectives.

12

Usually we avoid this difficulty by tacitly limit-ng the physical size of allowable test items, thus

aking the universe of test items finite.The same .problem arises with objectives at

igher cognitive levels. Again, tacit limitationsn the nature of test items are employed. Though

t is seldom done, it would be helpful if theseimitations were made explicit.

FeasibilityThis dimension refers to the extent to which

e, as teachers, are able to move students towardparticular objective. Refer again to the first

illustrative objective of the preceding section. Wenow by experience that we can teach most stu-ents, at the junior high school level, to transform

fractions into percents or decimals. Consequently,this objective is feasible for most students.

On the other hand, consider these objectives:The student should develop study skills which

will enable him to study mathematics independen tly and effectively.

Harding (1968), p. 101.Ideal of perfection as to logical structure,

precision of statement and of thought, logicalreasoning (as exemplified in the geometric dem-onstration).

Oakes (1965), p. 36.To discover new geometrical facts.Breslich (1930), p. 205.A gradual and increasing development of the

ability to manifest coherent, logical thinking ineveryday life situations.

joint Commission of MAA and NCTM(1940), p. 67.

The development of the habit and ideal ofpersistence.

Breslich (1930), p. 208.It is not at all clear that we know how to teach

our students to improve their performance withrespect to any of these objectives. Also, we havealready noted that none of the affective objectiveswhich have appeared in the literature have beendemonstrated to be feasible.6. Population

The dimension of population identifies the in-dividual or group of individuals for which theobjective is stated. The variation within popula-tion may be with respect to not only age level butalso social-economic status, mental maturity, fu-ture aspirations, mathematical background, re-gional location, etc.

13

In general, a set of objectives is intended forparticular population. For example, the NE(1894) Report was primarily for secondary education but several objectives or recommendationwere given for grammar school (6-13 years oage). The International Commission (1911) gava few precise objectives for kindergarten mathematics but was mostly concerned with the elenientary schools.- The UICSM's curriculum aneobjectives (Beberman, 1958) stated explicitly thattheir program was intended for the college bound

Population is often stipulated within the objettive itself. For example,

Intelligent guidance should guarantee thatevery high school pupil study mathematics according to his need and ability.

Pre-Nduction Courses in Mathematics (1943),p. 185.

Here the population addressed contains only one,the individual.

lir the other extreme, many objectives arewritten for a large population. For example,

Every high school pupil must be able to com-pute with assurance and skill....

Pre-Induction Courses iii Mathematics (1943),p. 115.

Between these two extremes lie several differentsubdivisions.

. . . for the mathematically less gifted pupilsin the ninth grade there is little straightforwardmathematics available beyond elementary in-struction in arithmetic and informal geometry,which should include guidance in the use offormulas, equations, graphs, and right triangles:

Harvard Committee (19.15), p. 163.For some, those with little or no mathemati-

cal background, this arithmetic plus certainessential topics from general mathematics suchas scale drawing, including elements of blue-print reading, numerical trigonometry, informalgeometry, and simple formulas and equations,will constitute the course.

Pre-Induction Courses in Mathematics (1943),p. 121.

The last two examples restrict the populationby ability in mathematics. The former concernsinnate ability, whereas the latter is more con-cerned with previous mathematics courses takenby the students.

One last example illustrates how some objec-

14

Ives are restricted in population by the futurespirations of the student.

The prospective candidate for admission tocollege certainly needs instruction in both alge-bra and demonstrative geometry.

Harvard Committee (1945), p. 167.

These examples only illustrate a few of theays in which population can be designated or

ulxlivided. What is important is that' the desig-ation and subdivision is very important to thecaning of the objective.

FormThis dimension has three values, Behavioral,

xpressive, and Pedagogical.Behavioralists believe that intended educational

hanges should be expressed in terms of mea-surable student behaviors. W James Popham de-

ribes a behavioral objective as follows:A satisfactory instructional objective must

describe an observable behavior of the learneror a product which is a consequence of learnerbehavior.

Popham (1969), p. 35.

As an example of a behavioral objective, con-sider the following objective:

GiVell a whole number between one and onehundred, the student will identify it as a primeor a composite number.

Instructional Objectives Exchange, grades 4-6(1968), p. 58.

There is some disagreement on the amount ofspecificity required. Sonic believe that the impor-tant outcomes of education are rather global andinarticulate a t tit tides, that concepts are developedradually as a byproduct of lessons, andthat the

important changes can rarely be observed overshort timespans. Those holding this view mightselect this objective:

The major role of mathematics in develop-ing desirable characteristics lies in the contribu-tions it can make to growth in the ability in-volved in reflective thinking or problem solving.

Progressive Education Association (1938),p. 59.

However, while it is harder to construct a goodtest of problem solving ability than it is to de-velop a test on prime and composite numbers, thesecond of these two objectives is no less behavioralthan the first.

15

. Eisner, a major proponent of expressive objetives, describes this type of objective as follows

An expressive objective describes an educational encounter: it identifies a situation iwhich children are to work, a problem witlwhich they are to cope, a task in which they arto engage; but it does not specify what from thaencounter, situation, problem, or task they arto learn.

Eisner (1969), p. 15.

An example of an expressive objective in mathematics is:

To expose the student to these ideas (concepts of problem formulation and mathematicamodeling) as broadening experiences. . . .

SMSG (1971), Chapter 4, teacher's commentart', p. 48.

Pedagogical objectives state how somethinshould be taught. Usually it is clear what the student should learn, but empirical evidence thathe method really works is very seldom providedHere are some examples:

Mathematics is by no means the only roa'to an appreciation of abstraction and logicastructures. But tactically these ends may be aproached most readily through mathematicsparticularly with the young.

Harvard Committee (1945), p. 162.A very large number of teachers would

well advised to tackle a greater variety of constructions than they are accustomed to, and tilay more stress on them.

Mathematical Association (1959), p. 75.

Comments on the DimensionsA. It is clear that while these dimensions can bedelineated separately, they must be consideredtogether in practice. We have already noted aninteraction between population and cognitivelevel. Addition of two threeigit numbers is acomputational task for a sixth grader but requiresanalysis of second grader. Population also inter-acts with content.

The goals:. . . Fundamental theorem of algebra, wind-

ing number, location of roots.Cambridge Conference (1963), p. 44,

even today would not be unreasonable for selectedhigh school seniors. However,. despite Bruner'sfamous statement (".. . any subject can be taught

16

ffectively in some intellectually honest form tony child at any stage of development!' Bruner

[1960], p. 33), it would be nonsense to advocatehis goal for first graders.

Feasibility probably interacts also with popu-ation. Abstract deductive reasoning, for example,oes not seem to be a feasible objective for ele-

nentary school students, but it is for many olderstudents.

And, of course, there' is no point in talkingabout the feasibility of an objective if we knowit cannot be verified, or even if we don't knowthat it can be verified.

B. The advocator of any .specific objective formathematics education should expect to be asked"why?". The answer to such a question begins withthe word "because" but can continue in two dif-ferent ways. One way would be along the linesof: "achievement of this objective is a good thingin itself:' The other would be along the lines of:"it is a prerequisite to learning another topic!'

Note that the first kind of answer expresses avalue judgment while the second makes a factualstatement. Now, when two individuals differ withrespect to a particular value statement, there isno rAional way of adjudicating, the matter andwe can only note that they differ. However, dif-ferences with respect to purportedly factual state-ments can be examined empirically.

Thus it appears that the statement of an ob-jective is incomplete unless it is accompanied bythe statement of its purpose, the answer as to whythe objective was selected.

Many of the sets of objectives that have beenpublished, and this is particularly true of someof the sets of behavioral objectives which haveappeared recently, are seriously deficient becausethey lack statements of purposes.* Impection ofthe individual objectives in these sets tirely yieldsone which is clearly intrinsically important, andfor many others it is hard to find important ob-jectives for which they are prerequisite.

An SIIISG Statement on Objectives inMathematics Education

As mentioned in the introduction, the impor-tance of the objectives of mathematics educationhas recently been substantially enhanced, The

A second objection to many of the presently available setsof behavioral objectives is that the cognitive levels ofknowledge and computation arc heavily overrepresented,while the levels of comprehension, application, and analysisare badly neglected.

17

SMSG Advisory Board hopes that those statinobjectives and also those who use statements oobjectives will keep the following principles inmind.

I. Statements of objectives should be hortatory.They should be taken seriously by teachers,curriculum workers, and textbook writers asimportant and realistic guidelines. Theyshould not be expressions of wishful think-ing.

II. On the other hand, statements of objectivesshould be taken as floors, not ceilings. If ateacher or a school can go beyond statedobjectives, so much the better.

III. If the statement of a particular objective isto he taken seriously, then the purpose of theobjective has to be made clear.. Further, aserious, relevant objective must be so clearlycharacterized as such as to be easily distin-guishable from a personal whim.

IV. If statements objectives are to be taken se-riously, then the objectives must be clearlyverifiable and feasible. It is not enough toknow that an objective has not been shownto be infeasible. Before it should be advo-cated, it should have been positively shownto be feasible (and verifiable)..

V. To be consonant with the above, we believethat all statements of mathematics educa-tional objectives should be put in terms ofstudent behavior. [The one exception is thatwe advocate a particular pedagogical objec-tive:

Teach Understanding of a mathematicalprocess before developing skill in the process.We believe there is enough empirical evi-dence in favor of this to make it a realisticobjective.]

VI. Also, to be in conformance with point IV,we advocate at present no affective objec-tives. There is no evidence available to showthat attitudes toward mathematics can bemanipulated, so such objectives are not, atpresent, feasible.

VII. None of the above should be taken as sug-gesting that we ignore goals which are, atthe moment, not feasible or not verifiable.Indeed, such goals indicate the most Cmpor-tant areas in which to concentrate our fu-ture research efforts.

18

BIBLIOGRAPHY

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INSTRUCTIONAL OBJECTIVESEXCHANGE (1968)

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21

SECTION IIMINIMUM GOALS FOR

MATEIFNE-t-rics EDUCATION

The School Mathematics Stud,, Group recentlycompleted a new mathematics cm riculum, Sec-ondary School Mathematics, for junior highschool students. A full description of this cur-riculum can be found in SMSG Newsletter No. 36.

At its final meeting, in April 1972, the SMSGAdvisory Board reviewed and approved this newcurriculum and decided that one aspect of it wasso important that it should be specifically calledto the attention of the mathematical community.in an SMSG Newsletter.

Secondary School Mathematics was designed toinclude those mathematical concepts, and onlythose, which all citizens should know in order tofunction effectively in our society. A list of theseconcepts is therefore SMSG's statement of theminimal goals of mathematics education. Thissection of this Newsletter provides a brief sum-mary of these minimal goals.

The writing team which produced this newcurriculum was asked to provide, for each chapter,reasons for including the chapter, or, in otherwords, an explanation of how the concepts in thechapter could contribute to effective functioningin our society. In response, the writers 'provided,for each chapter, one or more paragraphs listingthe objectives of the chapter and others statingthe purpose of the chapter.

These paragraphs, reproduced in what follos,provide our brief summary of the minimal goalsof mathematics. Before turning to them, however,some comments are necessary.

While the writers used the word "objectives:' itwas not at all in the sense in which the word isused in Section I of this Newsletter. The analysisof objectives in Section 1 was carried out after thewriters had finished their work. From the point ofview of that analysis, the writers' "objectives"could well be replaced by "goals:'

Nevertheless, some aspects of the analysis canbe applied to the writers' statements.

It has already been mentioned how the con,tent of Secondary School Mathematics was se-lected. The writers' statements, however, are muchtoo brief to be able to specify in any detail theemphasis an0 depth of coverage given to any par-ticular concept. Such information can only be ob-ftained from inspection of the actual text materials.

At the end of the teacher's commentary for each

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hapter is a set of illustrative test items. Inspectionf these will show that the writers expected stu-ents to be able to treat most mathematical con-epts in the chapter at all cognitive levels. Theseest items also demonstrate that the writers' goalsre verifiable.It is interesting to note that the writers did not

ist affective subjectives.As reported in Newsletter No. 36, Secondary

chool Mathematics underwent sa..iantial pilotesting and this curriculum is unquestionablyeasible, in its regular version for average andbove average students, and in its special editionor less able students.

Some of the writers' objectives are stated inerms of student behaviors, but a considerablemount of translation of the writers' objectivesnto behaviciral form would be needed before anympirical evalUation either of the curriculum orf teachers using it could be done.Finally, when the writers' statements were re-

iewed by the Advisory Board, some comments onpecific chapters were made and are included inhat follows. A few more general comments are

)laced at the end.

SUMMARY STATEMENTS OFTHE GOALS OF

SECONDARY SCHOOL MATHEMATICS

Chapter ISTRUCTURING SPACEPurpose

This introductory chapter contains several fun-damental ideas that are essential for the furtherstudy of mathematics. Specifically, the ground-work is laid for later chapters on geometry. Theidea of the coordinate line and coordinate planewill be encountered throughout the course. Inaddition, the "structing of space" provides-a basicexample of abstracting a mathematical modelfrom the physical world. The importance of thislatter idea is a continuing theme.

00c:divesSections 1 through 8: The student should un-

derstand that geometry is an idedl model of cer-tain aspects of the physical world; a hill appre-ciation of this is quite difficult. By explicit defi-nitions and characterization by properties, we tryto assure that various people's ideas (of plane, forinstance) are in essential agreement. These sec-tions are largely involved with definitions; thestudent should know them and be able to apply

23

them at least to the analysis of a given figure anto making a correct drawing of a figure havin,given properties.

Sectrons 9 through 12: The important pointthat by the coordinatization, every pointnamed by a number pair and every number painames a point. The ability to name given pointby their coordinates and to locate the point haing given coordinates is developed through etensive exercises and is the essential "skill:'

CommentsAdditional objectives to be included:

Knowledge of geometric facts.Applying these geometric facts to "real world'

problems.Learn to visualize 3-dimensional representation

learn to communicate ideas with regard to spatand be able to interpret geometric ideas witphysical world.

It is suggested that some statement might bmade to the effect that problem-solving still enterin everyone's needs and that problems requiringeometric sense (measurement problems, coordinate systems, angles, etc.) are in abundance in aplication.

Chapter 2FUNCTIONSPurpose

The function concept is certainly one of thmost significant ideas in mathematics. This firsexposure to functions is intended to be gentleThere are many examples of functions and ovarious ways that functions may be representedThe student is not expected to master the concept. As he progresses through the sequence thfunction idea will recur in almost every chapterSince physical situations often lead to functionand graphs, we pursue the idea of mathematicamodeling. The use of the terms input and outpuof a function serves as an introduction to Chapte3.

QbjectivesThe student should be familiar with the nota-

tion, language, and various means of representinfunctions. In particular, he should understandthat with each input corresponds exactly one out-put.

The student should be able to recognize simplefunctions in various forms. For reasonably simplefunctions, he should be able to find the outputgiven the input.

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CommentObjectives, second paragraph, first sentence:

Students should be able to recognize simple func-tions in various forms such as, tables, equations,diagrams, statements and graphs.

Chapter 3 INFORMAL ALGORITHMS ANDFLOW CHARTSPurpose

Algorithms are important in almost everybranch of mathematics. In developing an algo-rithm we must construct a detailed step-by-stepplan. Flow charting offers excellent insight intothe nature of algorithms and provides an intro-duction to a later chapter on computers and pro-gramming. Even divorced from its use with com-puters, the material on flow charts is valuable initself. Flow charts'are included wherever feasiblethroughout the course. The particular example ofthe division algorithm for integers is exploited inChapter 5 (Number Theory).

ObjectivesThe student should gain some familiarity with

the concept and use of flow charts and functionsof various boxes in a flow chart. He should under-stand that the processes used to find solutions ofproblems are often step-by-step procedures thatwe call algorithms. These procedures are par-ticularly useful in work with computers. The stu-dent should develop the skills necessary to expresssimple mathematical algorithms in flow chartform.

Chapter 4PROBLEM FORMULATIONPurpose

The primary goal of the chapter is to examinea bit more closely the topic of mathematical mod-eling. At the heart of applications of mathematicsis the process of finding an appropriate mathe-matical model of a real world situation. In factthis is often extremely difficult. We strive, in thischapter, to stimulate the student to realize thatreal world probleMs do not always resemble text-book problems. Chapter 4 is exploratory. Nospecific mathematical ideas are developed. Prob-lem formulation is stressed here, not problemsolving.

ObjectivesWe intend this chapter to be a soft-sell intro-

duction to the concepts of problem formulationand mathematical modeling. We want the student

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to be exposed to these ideas as broadening ex-periences and for this reason have not made de-mands on his competency. In presentation of thischapter, we encourage an informal approach witha lot of student participation in open-ended dis-cussions. The student may find this format un-familiar and may need extensive guidance at thestart with regard to how to approach the task ofproblem formulation. IVe include possible direc-tions along which he might be steered, but wehope that this "guidance" would not so structurehis thinking that he finds it necessary to fit histhinking into a mold. In other words, we appre-ciate that the student has imagination and do notwish to stifle it. For this reason, the "solutions"to the exercises consist only of suggested solutionsand should not be considered to be irrefutable.,

In order to tip the scale toward attaining theseobjectives, we provide more illustrative examplesand exercises than one might use for a week'sdiscussions. IVe hope that by doing so, the teachercan pick and choose examples that are likely tobe better-received according to the interests of theparticular class.

Our aim is to encourage open-endedness andcreativity as well as an enthusiastic attitude to-ward the ideas of problem formulation, ideaswhich in all likelihood, heretofore, :ire unfamiliarto the student. We have included no suggested testitems for this chapter. In fact, to promote the ac-quisition of this attitude toward problems, we ad-vocate a relaxed classroom atmosphere and we fur-ther advocate that NO TEST be,administered onthiS chapter!

CommentsStudents are given a situation and then they areanswer the question, "What are some mathe-

matical problems coming out of the situation?"

Chapter 5NUMBER THEORYPurpose

Students have had a good deal of experiencewith whole numbers. This chapter reviews somefamiliar concepts and develops., some extremelyimportant ideas including the statement of theFundamental Theorem of Arithmetic. Other con -cepts considered are those of the greatest commondivisor and the least common multiple. Althoughthese number theoretic results are interesting inthemselves, the primary motive in presentingthem is because they are necessary. tools in thetreatment of rational numbers, real numbers,

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polynomials, and factoring. The chapter containsthe first proof; includes proof that there is nolargest prime. The square theorem will be neededin Chapter 15. A summary of the whole numberproperties is included. The structure of the systemof integers follows in Chapter 6.

ObjectivesBy the end of the chapter, the student should

understand the division algorithm and the con-cept of divisibility. He should know the meaningof primes, how to find some primes, and how towrite a composite number as a product of primes.

CommentWe saw no test items for understanding the

division algorithm.

Chapter 6THE INTEGERSPurpose

This chapter is really preliminary to Chapter 7.Rather than combine the problems of negativenumbers with the manipulations of rationals, thenegative integers are introduced here. The exten-sion of the number line, the order relation, addi-tion, subtraction, and multiplication using nega-tive numbers are illustrated using negative in-tegers. The important structural difference be-tween the system of whole numbers and the systemof integers may be summarized by the statement:"If a and b are any integers, then there is exactlyone integer x such that a x = The com-parable statement for whole numbers is false.

ObjectivesThe system of integers, in itself, is not particu-

arty interesting nor important for the student.ur goal is to separate the problems of manipulat-

ng "negative" numbers from the problems ofanipulating rational numbers (Chapter 7). Once

ie student has learned, for example, that(-2) (9) = 18

can handle

first writing(- (-1)

(- 3)(1)= (T) (1)then he can concentrate on the problem of

tiplying the positive rationals 2 and 93 4

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CommentsAt end of objectives paragraph, add: "Given

any two integers, the student can find the sum,difference, product, and, if it exists, the quotient!'

The board disagrees with the first sentence inthe Objectives paragraph.

The topic of absolute value provides a new wayof illustrating the concept of function.

Chapter 7THE RATIONAL NUMBERSPurpose

It is important that the student acquire skillin handling rational numbers. In this chapter thetechniques for arithmetic operations with rationalnumbers are developed. Further, we wish to lay thefoundations for later chapters. In Chapter 9 weshall solve equations of the form ax b = 0 andrational numbers are needed. In Chapter 15, thestudent learns that although the rational numbersare dense they are not sufficient to "fill up" thenumber line nor to solve such equations asx2 2 = 0.

ObjectivesA first objective is to review the ordinary arith-

metic of the nonnegative rationals. The usualrules for addition and multiplication are ex-plained. Secondly, we wish the student to carryover his knowledge of manipulation of integers(positive, negative, and zero) in order to applythem to the rational numbers. Finally, we wishthe student to recognize two fundamental differ-ences betwen the structure of the rational num,hers and the ,structure of the integers. Most im-portant is the existence of the multiplicative in-verse (reciprocal) of a non-zero rational. In addi-tion, the rational numbers are dense; this prop-erty is not shared by the integers.

CommentAdd to objectives: to recall skills with non -

negative rationals and to extend them to allrationals.

Chapter 8CONGRUENCEPurpose

The notion of congriience (of plane figures) isnecessary in order for us to pursue the study ofgeometry which was begun in Chapter. 1 and iscontinued in several of the later chapters. Con-gruence is also used as a basis for measurement(Chapter 12). Apart from its relation to otherchapters, Chapter 8 is important for other reasons.The student is given some experience with proofs.

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Further, there are many practical applications ofcongruence.

ObjectivesIt is anticipated that the student will develop

an understanding of the general nature of con-gruence. In particular, the student should be ableto recognize when SAS, ASA, SSS situations occurfor triangles and be able to draw correct conclu-sions.

Much of the geometric vocabulary introducedhere is new. It will be reviewed and reinforced insubsequent chapters. For example, the subject ofperpendicularity is taken up in more detail inChapter 13. The student should be able to use thevocabulary. He should recognize, say, a quadri-lateral. without necessarily being able to give a_formal definition.

This is the student's first exposure to geometricproof. Obviously complete mastery of the natureof proof is not expected. It is sufficient that the-,:tp;-,Ient understand the proofs of the isosceles tri-angle theorems. More proofs will be encounteredlater.

CommentsThe last sentence of the first paragraph, "Fur-

ther there are many practical applications of con-gruence:' realty should lead the paragraph.

The kind of proof referred to in the statementshould be described. Perhaps, ". . . using logicalreasoning to establish facts (relationships) ingeometry, ..."

Change "geometric proof" to "geometric rea-soning" in first sentence of last paragraph.

Chapter 9EQUATIONS ANDINEQUALITIESPlopose

This chapter presents a systematic developmentof methods for solving some simple first degreeequations and inequalities in one variable. It isnot intended here to develop complete masteryof the process of finding solution sets of equationsand inequalities. This topic will be spiraled con-sistently throughout the remainder of the cur-riculum. This chapter also provides an analysis

' of how these equations and inequalities imy beformulated as models of problem situations.

ObjectivesThe central objectives of this chapter are(1) to translate English phrases into mathe-

matical expressions and to develop equa-

29 .

Lions as models of certain problem situa-tions,

(2) to solve equations of the form ax b = cand ax b =_- cx d using the additionand multiplication properties of equalityand the field properties of the rationals,

(3) to use a graphical approach in solving theseequations,

(4) to develop inequalities as models of certainproblem situations,

(5) to solve inequalities in a way similar tosolving equations,

(6) to review properties of integers and ration-als and to increase computational skill intheir use.

Chapter 10DECIMAL REPRESENTATIONFOR RATIONAL NUMBERSPurpose

In this chapter we revisit the rational numbers.In Chapter 7 (The Rational Numbers), the em-phasis was on extending the number system andon the operations with rational numbers ex-pressed as fractions. Here we consider the rationalnumbers as ratios, as percents, and as repeating(or terminating) decimals. Rounding off and com-puting with rational numbers expressed as deci-mals is reviewed and explained in the last section.These topics will be used frequently in later chap-ters, including Chapter 11 (Probability) and Chap-ter 12 (Measurement). An example is given of aninfinite decimal that does not represent a rationalnumber. This idea will be extended in Chapter15 (The Real Numbers).

ObjectivesAfter studying this chapter, students should be

able to express a ratio as a rational number, ex-press simple fractions as percents, write a decimalin expanded notation, find the decimal represen-tation of a rational number and the simplest frac-tional name of a terminating or repeating deci-mal, and round off and compute with decimals.He should know that there are infinite decimalswhich are not rational numbers.

CommentA realization that ratios are useful should be

included among the objectives.

Chapter 11PROBABILITYPurpose

Some understanding of probability and statis-

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tics is essential for the educated citizen in modernsociety. This chapter introduces some fundamen-tal ideas about probability. Probability theory isa requisite for the techniques of statistical analysisand statistical inference that play so large a rolein industry, government, economics, social scienceand all branches of physical and biological sci-ence. In Chapter 24, some elementary statisticalanalysis is presented. Some basic tools of statistical.inference are developed in Chapter 28, using theideas of Chapters 11 and 24.

0 qui InesWe hope to leave with the student an awareness

that. many situations or experiments lead to resultsthat are uncertain. In this introduction to proba-bility. we attempt to quantify the degree of un-certainty for less complex situations. This quan-tification is accomplished by assigning real num-bers from 0 to 1 to the outcomes.

The student should be able to use most of thevocabulary introduced in the chapter. It is notexpected that he would be able to give formaldefinitions. Uncertainty, equally likely, fair game,outcome, event, probability, and probability func-tion are all terms the student should be able touse readily. For clear-cut situations, he should beable to describe a se,. of outcomes, assign proba-bilities. and to find the probability of A U B, andA rI B where A and B are subsets of the set ofoutcomes.

Formal definitions are necessary for an under-standing of: mutually exclusive events, comple-mentary events, and independent events.Chapter 12MEASUREMENTPurpose

The great majority of applications of mathe-matics, whether arithmetical, algebraic, geometric,statistical, or analytical are rooted on the idea ofmeasurement. Historically, of course, mathematicsbegan with need to measure and to record mea-surements of various kinds. In this chapter, wedevelop basic ideas of the measurement of lengthand of angles, based on the notion of congruencefrom Chapter 8. We deal both with theoreticalonsiderat ions and properties of measures as wells with the practical problems of obtaining, re-orting, and interpreting measurements of physi-al objects. These ideas will be used in Chapter 14Similarity) and elsewhere in the sequence. Inhapter 16, we shall develop measures of area andlame and introduce the Idea of indirect mea-rement.

31

ObjectivesThe student should understand that any seg-

ment can be used as a unit segment. He shouldbe familiar with the standard units of length inthe British and metric systems and be able to con-vert from any unit to any other if he is given theconversion function. He should not be expectedto memorize the conversion functions.

He should 'know and be able to use the formulasfor calculating the perimeter of a polygon and thecircumference of a circle.

He should understand the definition of a degreeand the terms "acute angle': "right angle:' and"obtuse angle" although he may not be able todefine them carefully. He should know that thesum of the measure in degrees of the angles of

iany triangle is 180.Most importantly he should understand that

physical measurements arc approximate and havesome understanding of the reason for giving theprecision of a measurement. He should under-stand how to use a protractor to measure anglesin degrees.

CommentInclude the objective of being able to convert

within a particular system from one measurementto another; for example, a particular measurementfrom feet to inches, etc.

Chapter 13 PERPENDICULARSAND PARALLELS (I)Purpose

The theory of perpendiculars and parallels isnot only basic for the study of Euclidean geometry,but in fact, most practical applications of geometry involve this theory. In particular, importantresults concerning rectangles and parallelogramsare developed in this chapter. From the theoreticalpoint of view, work on parallels and perpendicu-lars offers more practice for the student in under-standing and designing proofs. Finally, the theo-rems of this chapter are essential for further geo-metric study such as that presented in Chapters14 and 17.

ObjectivesThe student should understand the content and

some consequences of the theorems listed in thechapter summary although he may not be able tostate each theorem' precisely. In particular, heshould understand several ways of deciding if twolines are parallel, be able to identify alternate in-

32

terior and corresponding angles, be able to define"rectangle: "rhombus': and "parallelogram," andknow properties of and relationships betweenthese quadrilaterals. He should be able to con-struct parallel and perpendicular lines, a rec-tangle, a square, a rhombus (from Chapter 8), anda parallelogram. He should learn more aboutproving theorems, be able to supply reasons orstatements in proofs, and be able to construct asimple proof when given the method to follow.

He should be aware that the theorem that thesum of the degree measures of the angles of anytriangle is 180 depends upon the Parallel Postu-late and should be able to use this theorem. Heshould be able to follow the use of parallels inthe Eratosthenes derivation of the earth's circum-ference. The student should not be expected toproduce original proofs (unless they are verysimple and straightforward) or to write precisestatements of complicated theorems.

CommentPenultimate sentence, first paragraph of Ob-

jectives: ". . and a parallelogram, with possibly avariety of tools:' Ultimate sentence: ". . . supplyreasons or statements in geometric reasoning orarguments):'

Chapter 14SIMILARITYPurpose

Most students have a strong intuitive recogni-tion that many objects appear to have the "sameshape:' The purpose of this chapter is to examinethe mathematical basis for this property of "sameshapeness:' Specifically we reduce it to the con-gruence of corresponding angles, discussed inChapter 8 (Congruence), and the equality of ra-tios of lengths of corresponding line segments.The measurement of length and of angles was de-veloped in Chapter 12 (Measurement). The chap-ter draws on what the student has learned aboutparallel lines and transversals in Chapter 13. Anattempt has been made to keep in the foregroundsome of the many applications of similarity; it isthe basis of practically all surveying. Use is madeof similarity to obtain a particularly easy proof ofthe Pythagorean Theorem. An introduction totrigonometric functions at the end of the chapterollows naturally from the study of similarity.

6/calvesBy the end of the chapter, the student shouldfamiliar with the conditions for similarity of

tangles. For example, he should recognize that

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two triangles are similar if each has a 50° angleand a 75° angle. It is expected that he should beable to solve simple problems involving similartriangles. He should understand the statement ofthe Pythagorean Theorem and know how to ap-ply it to problems involving right triangles, suchas finding the length of one side of a right trianglewhen the lengths of the other two sides are known.As for the three trigonometric functions, the stu-dent should remember how they are defined interms of adjacent side. opposite side, and hypote-nuse, and be able to use these functions (with helpof the table of values) to solve problems involvingright triangles.

Chapter 15THE REAL NUMBERSPurpose

From the time of the Pythagoreans to the pres-ent day, no topic in mathematics has played suchan important role as the understanding of the realnumber system. The concepts involved are difficultand challenging. From Chapters 7, 10, and 12, werecall that there are points on the number lintthat cannot be identified With rational numbers.In this chapter. we develop the idea that we may"complete" the number line by introducing irra-tional numbers. Many irrational numbers arisefrom geometric considerations such as the lengthof the diagonal of the unit square. Therefore, webuild on the material of the earlier chapters ongeometry. The qfiestion of rational approxima-tions is discussed, with emphasis on square roots.

ObjectivesAn objective of this chapter is to "complete"

the numbers on the number line by introducingthe idea of irrational numbers. Geometric lengthsexist as real "lengths" even though the set of ra-tional numbers is not adequate to specify theexact lengths of many segments. The concepts oflength and distance naturally lead to square rootsand we include as an objective some facility inoperating with square mots, manipulating radi-cals. and obtaining rational approximations.Other objectives include better understanding ofthe structure of the number systems through theextensions from the natural numbers to the realnumbers.

The following is a suggestion for further read-ing on the subject for teachers and students. Theseare volumes in the series of The New Mathemati-cal Library, SMSG:

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Numbers: Rational and Irrational by IvanNiven

The Lore of Large Numbers by Philip DavisUses of Infinity by Leo Zippin

Chapter 16SECOND LOOK ATMEASUREMENTPurpose

This chapter continues the study of measure-ment begun in Chapter 12. It also relies on thediscussio% of congruence in Chapter 8, decimalrepresentation in Chapter 10, properties of rec-tangles in Chapter 13, the Pythagorean Theoremin Chapter 14, and computation with real num-bers in Chapter 15. Many of the formulas for areaand volume are developed by the students iii. exer-cises. The discussion of precision in Chapter 12is extended to computation with ineasureruznts(approximate numbers) and the hazards involve'.Other kinds of direct measurement and some ex-amples of indirect measurements that studentswill encounter in such courses as science and shopare introduced.

ObjectivesThe student should know and be able to use the

formulas for the area of a rectangle, triangle, par-allelogram, trapezoid, and circle. He should un-derstand and be able to use the formulas for thearea of a rhombus (in terms of its diagonals), aring, and a sector of a circle if he is given the for-mula or is reminded of its derivation. He shouldbe able to derive similar formulas if they arepresented to him in a step-by-step approach.

He should know and be able to use the for-ulas for the volume of a box and a sphere. He

hould know that the volume of many solids,ncluding triangular prisms and cylinders, is theroduct of the area of the base and the height ofe solid. He should be familiar with standardnits of area and volume. He should be able tose the definitions of surface area of each solidit should not be expected to memorize the for-ulas.He should be able to make Feneralizationsout hew area or volume changes when theear dimensions are doubled, tripled, or halved.should be aware that error can be compounded

en he computes with numbers that are ap-ximations. He should understand that thereother kinds of measurements (indirect) whichnot have 'the properties that length, area, andme do.

35

Most importantly, this chapter should contrib-ute to his understanding of mathematical deri-vations and of how mathematics is applied to thereal world.

CommentsAdditional objective: Understanding of units

in the sense that the particular kind of unit usedfor measurements is important; that is, for in-stance, square. inches are used for areas expressedin inches; that conversions may be made, butinches .are not mixed up with feet in the compu-tations.

Second sentence in fist paragraph, Objectives,suggest changing to: He should understand and beable to apply these formulas to other areas con -,strutted from other basic areas.

Oapter 17PERPENDICULARS ANDPARALLELS (II)Purpose

A major purpose of this chapter is tc derivemethods of arriving at conjectures... to recognizethat conjectures require proof, and to prove someof them. The methods of arriving :a conjecturesinclude paper folding and asking whether theo-rems about parallel and perpendicular lines in aplane can be generalized to space. Another pur-pose of the last three sections is to develop thestudent's ability to visualize 3-dimensional figuresand relations in space. This chapter builds on andextends the geometry developed in Chapters I,8, and 13. Chapters 13 and 17 together containthe usual theorems on parallels and perpendicu-lars from Euclidean .geometry which have manyimportant applications in the real world.

ObjectivesThe student should understand the meaning of

such terms as "altitude," "angle bisector; "perpen-dicular bisector; "median:' "tangent to a circler"dihedral angler "plane angle of a dihedral angler

. and "mutually perpendicular planes:' He shouldbe able to construct by paper folding the variouslines associated with a triangle, such as the me-dians, and see how compass and straightedge con-structions are related to paper folding "construc-tions:' He should be able to construct a tangentto a circle at a point of tangency or from an ex-ternal point and to construct a rectangle having agiven side and a given diagonal (provided thediagonal is longer than the side). He should beable to prove simple theorems about medians, etc.,of a triangle, use the hypotenuse-leg theorem, un-

36

derstand and explain the tangent constructions,and know that any angle inscribed in a semicircleis a right angle. He should be able to visualizesuch figures as parallel, skew, and perpendicularlines in space, parallel and perpendicular planes,a line perpendicular to or parallel to a plane, andthe set of points equidistant from the end-pointsof a line segment in space. He should be able tothink about whether a theorem in 2-dimensionalspace can be generalized to a theorem in 3-dimen-sional space.

Chapter 18COORDINATE GEOMETRYPurpose

This chapter brings together the previous de-velopment of geometry, of functions, and the alge-braic development of number systems, and laysthe foundation for the study of a mathematicalsystem called analytical geometry. Therefore, thischapter is a key chapter in the grade 7 throughgrade 9 sequence.

The purposes of this chapter are(-l-)to- introduce the students to the way in

which certain geometric ideas may be for-mulated and expressed algebraically; .

(2) to indicate the power of these algebraic for-mulations in analyzing some mathematicalideas.

The following background, developed in earlierchapters, is assumed:

(I) familiarity with coordinates of points on aline and coordinates of points in a planeand how to plot them;

(2) some understanding of the function con-cept;

13) ability to graph linear functions;(4) understanding of perpendicularity and par-

allelism, and of definitions and propertiesof certain geometric figures;

(5) ability to transform equations from one .'form to another.

ObjectivesThe student should be able to(I) use meaningfully the distance and midpoint

formulas in 1 -, 2-, and 3-dimensional co-ordinate systems,

(2) describe algebraically, using set builder no-tation, certain geometric figures ire each of

. the coordinate systems, and conversely, todescribe geometrically the solution sets of

37

points of certain equations,(3) determine slopes of lines and to relate slope

to parallel and perpendicular lines,(4) develop coordinate proofs of certain proper-

ties v.: some geometric figures.

Chapter 19PROBLEM SOLVINGPurpose

All of mathematics is concerned with problemsolving. Sonie problems are theoretical and manyare "practical:' Problems of various types occur,obviously, throughout secondary school mathe-matics. in each chapter, we have been concernedwith specific techniques developed in the chapter.There are, however, certain general strategies andmethods that are useful in all types of problems.The major purposes of Chapter 19 are to directthe student's attention to some of these strategiesand techniques. The formulation of a mathemati-cal model for a given "real" problem, the analysisof this model, and the interpretation of the analy-

____sis_in_iermis_of_the-originaLeroblem_are_stresseHopefully, the practice provided by the.examplesand exercises of this chapter will be reinforcedthroughout the student's pursuit of his mathe,matical studies.

Objectives(a) lb provide the student with a variety of

strategies for problem r,;:i ving.(b) To develop some flexibility in the student's

approach to problem solving.(c) lb develop techniques for using geometric

. representations as a way of developing newinformation about a given situation.

(d) To develop some skill in using a tabular ar-rangement, of given and derived informa-tion., to help solve problems.

(e) To develop, for the student, a better under-standing of a problem by teaching him. tomake numerical estimates and testing theirkin the actual problem.

Chapter. 20SOLUTION SETS OFMATHEMATICAL SENTENCESPurpose

If there is one single skill which is the most--portant for the application's of mathematics,

the ability to solve simple algebraic equations(and inequalities). This ability is so fundamental.that it occurs in almost every chapter of Secondary.School Mathematics. In Chapter9, Are began the

38

formal development of techniques of finding so-lution sets. Chapter 20 reviews and extends thesetechniques. The stress is on the idea of deriving,,from a given open sentence, a chain of equivalentsentences using the properties of the real numbersystem. The final sentence in the chain should beone whose solution set is obviouS. Incorporatedin this chapter is some practice on simple manipu-lation with polynomials and rational fractions(although these terms are not used). While theemphasis is our linear equations and inequalities,some Isvork on quadratic polynomials lays thefoundation for Chapter 23 (Quadratic Functions).In addition, the skills developed in Chapter 20are, of course. fundamental for Chapter 25 (Sys-tems of Sentences in Two Variables).

ObjectivesWe are concerned with equations and inequali-

ties which, after simplification, involve only thefirst power of a single variable. The studentshould be able to 'solve such equations and in-equalities efficiently and_accurately,It-is-also-expCciF(111--iat the student will be able to handlecer-tain types of factoring. He should be able to solveequations winch are written as a product of linearfactors equal to 0.

Additionally, the student will gain experiencewith "translating" verbal problems into mathe-matical language and with solving the resultingsentences.

Com roodhi the first sentence of the Objectives statement,

"simplification" should be replaced by "someanalysis:'

Chapter 21RIGID MOTIONS AND.VECTORSPurpose

Euclidean geometry is, in essence, the study ofthe properties of figures which are preserved un-der a certain set of transformations. These trans-formations are those which are discussed in thischaptertranslations, rotations, and reflections.Thus, Chapter 21 is of primary importance for anUnderstanding of Euclidean geometry. In Chapter8 (Congruence). the student had experience inusing a tracing sheet. The process of moving aracing sheet from one position to another sugests the general idea of a rigid motion. We nowevelop the idea .more carefully. The importantotion of symmetry rises froM considering retie&ons. Coordinate geometry.(Chapter 18) is used

39

to express certain rigid motions in terms of a"change of coordinates:' This work is exploitedin Chapter 23 (Quadratic Functions). Transla-tions are representable by vectors, so a brief in-troduction to two-dimensional vectors is pre-sented. Applications are made to some of theusual problems involving navigation, equilibriumof forces, as well as to geometric proofs.

ObjectivesThe important items in this chapter are much

more the ideas than any particular operationalskills.(a) Understandings expected

The meaning of a rigid motion, in particularof translation, reflection, and rotation.The meaning of symmetry and axis of sym-metry.The composition of two rigid motions.That usually the composition of two rigid mo-tions depends on the order in which they are

-- taken.That for two translations the composition doesnot depend on the order.That a translation can be described by an ar-row.That a reflection is determined by its axis.That the coordinate plane can often be usedto. describe rigid motions.The association of vectors in a plane mithtranslations.The meaning of a vector sum.

(b) Performance expectedIf a rigid motion is suitably described, to findthe image of a given figure under this motion.For a given figure to decide if a given line isan axis of symmetry.To find all axes of symmetry for a simplefigure.In simple cases, given two congruent figures, todescribe explicitly a sequence of one or moretranslations, reflections, and rotations mappingone figure onto the other.Given a point in the coordinate place and itsimage under a translation, to find the imageof any other point. Given a point in _the _co,ordinate plane, to find its image under reflec-tion in the X-axis or in the Y-axis.TO solve simple problems involving velocitiesand forces making use of vector methods.

40

Use vectors to prove certain geometric theo-rems.

Chapter 22COMPUTERS ANDPROGRAMMINGPurpose

There is no need to stress the increasing im-portance of computers in modern society. Build-ing heavily on the flow charting work of Chapter3, we introduce the student to the elements of adigital computer, an assembly language, and tothe basic language. Use is made of algorithms de-veloped in earlier chapters to provide illustrativematerial. The chapter should .provide a minimal?understanding of the capacities_ and limitationsof computers. Students who wish to learn moreabout computers and computer linguage shouldfind this chapter a usefintroduction to pro-gramming.

Objectives;3tuclents should gain a better understanding

pf-tow-cii-rn-mtterrivoilcaTidaliFinIEW program-mers :41 making computers work by studying thiz.,chapter. If they have the opportunity to run someof their programs, they will gain much greaterinsight into the challenges of programming. Byanalyzing how a computer evaluates expressions,Students learn more about mathematical syinbo-lism and algorithms. Such algorithms as "divideand average" and the solution of the general lin-ear equation in one variable should become moremeaningful. Programming may motivate studentsto continue studying mathematics.

Chapter 23QUADRATIC FUNCTIONSPurpose

This chapter provides a systematic analysis ofquadratic functions and a formalization of meth-ods of solving quadratic equations. Extensive useis made of previously studied' concepts. In par-ticular, we build upon the study, of linear func-tions, upon coordinate geometry and upon theearly ideas about finding the solution sets ofmathematical sentences. The skills developed inthis chapter are useful in the later study of ex-ponential and logarithmic functions (Chapter 26).

ObjectivesThe student should be able to'(a) identify the essential characteristics of the

graph of a quadratic function when the. function is.expressed in standard form;

41

(b) solve quadratic equations by the method ofcompleting the square;

(c) state the necessary restrictions on the vari-able in equations involving square roots,absolute values, and rational expressionswhich lead to quadratic equations, and tosolve these equations.

CommentThe Board is not convinced that everyone needs

this much exposure to quadratic functions.

Chapter 24STATISTICSPurpose

The increasing reliance on statistical methodsin the decision making process in many branchesof human activity makes it essential that the edu-cated citizen have some familiarity with statisticaltechniques. In this ch..pter we attempt to preset.'some elementary ideas about organizing numericaldata and of reporting summary descriptions of th

dataWe_concentrate_on_the_use_of_theinean_andthe standard deviation as two numbers whichsummarize information about a collection of.nu-merical data.

Statistical methods are most often used to reachdecisions or to make predictions. Such decisionsand predictions involve probabilistic considera-tions. In Chapter 28 we shall,. bring together themethods of Chapter 24 and the probability theoryof Chapter II.

ObjectivesWe expect the student to understand how to

construct and understand simple tables andgraphs. In a similar way, he should understandthat the mean and standard deviation are usefulin reporting a "summary" of information abouta distribution. The level of expected competencyin calculation is shown by the nature of the exer-cises. The material on the use of the normal curve(Section 24-7) is preliminary. It will be reviewedand reemphasized in Chapter 28.

Chapter 25SYSTEMS OF SENTENCESIN TWO VARIABLESPurpose

All the machinery of problem formulation, con-struction of mathematical models, solution ofequations and inequalities, and coordinate geom-etry is brought together in this chapter. Challeng-ing problem situations are presented whose solu-tions involve the techniques of linear program-

42

ing. In order to solve such problems, the studentled to consider systems of linear equations andequalities. Graphical and algebraic methods oflution are presented. The algebraic argument is

on the idea of equivalent systems. The simi-r idea of equivalent sentences has been care-Ily developed in Chapters 9 and 20.

bjectivesThe student should become capable of solvingsystem of linear equations

-Ax + By 4- C = 0Dx + Ey + F=0,

It is expected that the student should be able toderstand and to use both the algebraic methods

f "elimination" and "substitution") as well asehical methods.

Similarly, the student should understand thate solution set of a linear inequality such as

+ By + C < 0 consists of the points in aIf-plane. He should be able to graph the solu-

ket-oLsysterri-oUtwo-ormorelinear mequali-es.We anticipate that the student will be able to

these techniques to arrive at the solution of anementary linear programming problem.

hapter 26EXPONENTS ANDOGARITHMSrose

A major purpose of this chapter is to developnipuJatory and computational skills. Using in-rs as exponents leads to the useful scientific

tation. The introduction of rational numbers asponents is of assistance in working with radicals.Simple arithmetic can be done by "longhand"ethocls; extensive calculations can be performedciently by computers. Logarithms are the major

in the middle ground for computations thattoo laborious for longhand and too simple to

stify expensive computer time.The development of the material in this chapterof interest in itself. The student discovers thate basic idea (positive integral exponents) cand, step-by-step, to the much more sophisticatedcepts of exponential and logarithmic func-

tis.

jectiveshis chapter provides the student's first expo-to a large number of new ideas, symbols, andpiques. It is too much to expect complete

airy. of them all. If the student can -handle

Use vectors to prove certain geometric theo-rems.

Chapter 22COMPUTERS ANDPROGRAMMINGPurpose

There is no need to stress the increasing im-portance of computers in modern society. Build-ing heavily on the flow charting work of Chapter3, we introduce the student to the elements of adigital computer, an assembly language, and tothe basic language. Use is made of algorithms de-veloped in earlier chapters to provide illustrativematerial. The chapter should.provide a minimalunderstanding of the capacities and limitationsof computers. Students who wish to learn moreabout computers and computer linguage shouldfind this chapter a useful- introduction to pro-gramming.

ObjectivesStudents should aina_betteliL-understanding

users work and the role of program-mers in making computers work by4studying thischapter. If they have the opportunity to run someof their programs, they will gain much greaterinsight into the challenges of programming. Byanalyzing itow a computer evaluates expressions,students learn more about mathematical syMbo-lism and algorithms. Such algorithms as "divideand average' and the solution of the general lin-ear equation in one variable should become moremeaningful. Programming may motivate studentsto continue studying mathematics.

Chapter 23QUADRATIC FUNCTIONSPurpose

This chapter provides a systematic analysis ofquadratic functions and a formalization of meth-ods of solving quadratic equations. Extensive use's made of previously studied concepts. In par-icular, we build upon the study of linear func-ions, upon coordinate geometry, and Upon-theany ideas about finding the solution sets ofathematical sentences. The skills developed inis chapter are useful in the later study of ex-

entail and logarithmic functions (Chapter 26).

bjectivesThe student should be able to.(a) identify the essential characteristics of the

graph of a quadratic function when thefunction is expressed in standard form;

41

(b) solve quadratic equations by the method ofcompleting the square;

(c) state the necessary restrictions on the vari-able in equations involving square roots,absolute values, and rational expressionswhich lead to quadratic equations, and tosolve these equations.

CornmentThe Board 'tot convinced that everyone needs

this much exposure to quadratic functions.

Chapter 24STATISTICSPurpose

The increasing reliance on statistical methodsin the decision making process in many branchesof human activity makes it essential that the educated citizen have some familiarity with statisticaltechniques. In this chapter we attempt to presentsome elementary ideas about organizing numericaldata and of reporting summary descriptions of the

a,Wc-concentrate-on--tlic-use_oLthe_mean_athe standard deviation as two numbers whichsummarize information about a collection of nu-merical data.

Statistical methods are most often used to reachdecisions or to make predictions. Such decisionand predictions involve probabilistic considera-tions. In Chapter 28 we shall bring together themethods of Chapter 24 and the probability theoryof Chapter 11.

ObjectivesWe expect the student to understand how to

construct and understand simple tables andgraphs. In a similar way, he should understandthat the mean and standard deviation are usefulin reporting a "summary" of information abouta distribution. The level of expected competencyin calculation is shown by the nature of the exer-cises. The material on the use of the normal curve(Section 24-7) is preliminary. It will be reviewedand reemphasized in Chapter 28.

Chapter 25SYSTEMS OF SENTENCESIN TWO VARIABLESPurpose

.All the machinery of problem formulation, con-struction of mathematical models, solution ofequations and inequalities, and coordinate geom-etry is brought together in this chapter. Challeng-ing problem situations are presented whose solu-tions involve the techniques of linear program-

42

I

ng. In order to solve such problems, the studentled to consider systems of linear equations andequalities. Graphical and algebraic methods oflution are presented. The algebraic argument issed on the idea of equivalent systems. The simi-

r idea of equivalent sentences has been care-illy developed in Chapters 9 and 20.

bjectivesThe student should become capable of solvingsystem of linear equations

Ax -I- By ± C = 0Dx + Ey ± F = O.

It is expected that the student should be able toderstand and to use both the algebraic methods"elimination" and "substitution") as well as

aphical methods.Similarly, the student should understand thate solution set of a linear inequality such asx By + C < 0 consists of the points in aIf-plane. He should be able to graph the solu--set-of-system-of-two-orrinear inequali-

es.We anticipate that the student will be able to

these techniques to arrive at the solution of anementary linear programming problem.

hapter 26EXPONENTS ANDOGARITHMSurposeA major purpose of this chapter is to developanipulatory and computational skills. Using fin-gers as exponents leads to the useful scientificcation: The introdiiction of rational numbeia asponents is of assistance in working with radicals.Simple arithmetic can be done by "longhand"ethods; extensive calculations can be performedciently by computers. Logarithms are the majorI in the middle ground for computations that

e too laborious for longhand and too simple tostify expensive computer time.The development of the material in this chapterof interest in itself. The student discovers thate basic idea (positive integral exponents) canad, step-by-step, to the much more sophisticatedncepts of exponential and logarithmic func-ns.

bjectivesThis chapter provides the student's first expo-re to a large number of new ideas, symbols, andhniques. It is too much to expect complete

astery of them all. If the student can handle

43

most of the exercises, this is sufficient. Specificawe hope the student acquires familiarity with

(1) the definition of aK for x an integer,(2) the interpretation of awq as N/777(3) exp, and logs as functions,(4) common logarithms.This familiarity should be sufficient so that

student is able to(1) manipulate Freely with integral exponen

including the ability to use scientific notion:

(2) simplify ordinary expressions involvingtional exponents or radicals;

(3) use a table of mantissas with sufficient skto apply, logarithmic techniques to anmetic computations.

Only a beginner's level of skill is anticipatFor example, it is not planned that the studentable to interpolate. It is expected that he wouhave to proceed very slowly in "simplifying"

tremeimessy-expressimg.TOnce again, thwhofor degree of skill in manipulation is indicatedthe exercises.

The section on the slide rule is strictly int-1tictory and no practical use of a slide rule isveloped.

Corn Gien!The second sentence of the second paragra

provides an illustration of the speed with whitechnology progresses. Less than a year after it wwritten, small, inexpensive, battery powered eltronic calculators were available to handle t"middle ground for computations that, are tlaborious for longhand and too simple to just'expensive computer time:'

Chapter 27LOGICPurpose

Logical thinking is inherent in all of mat=tics. From this point of view, a careful discsion of logical principles might have come earl'in the sequence. Until the student acquires a gdeal. of mathematical experience, however, itdifficult to illustrate the need for carefully orgnixed methods of deduction. We can in this chater discuss the logical framework which under!'the mathematical proofs to which, he hasexposed. As illustrations, we re-examine certaproofs used in preceding chapters. Both diproof and proof by contradiction are discussed

44

me detail. The methods of proof discussed inis chapter will be refined and utilized in futureurses.

bjeciivesAt the end of this chapter, the student shouldderstand and be able to correctly use the terms:

Simple statement, compound statement, ne-gation, conjunction, disjunction, conditional,biconditional, logically equivalent, converse,contrapositive, direct proof, proof by contra-diction, quantifier.

fe should be able to symbolize statements, con-ruct truth tables and use them to determineigical equivalence. He should be able to con-ruct a simple direct proof in either logical or

thematical form and complete a proof by eon-fiction. He should be able to state the nega-of a quantified statement, using Euler (ha-s as an aid if he wishes.

mentiraer- Purpose, include the notion that logical'liking is useful outside mathematics. Emphasize

necessity for precision in use of language.

pter 28PROBABILITY ANDATISTICS

ser ma or goal is to lead the student to somestanding of statistical results which are soreported. We frequeny encounter suchents as "the students of our school per-

significantly better than the average of allts in the school system;'. "the opinion poll.

voters shows that candidate A will be, or "the new medicine has no significanton the recovery rate of patients:' In this

we discuss the mathematical methods andts that are used in reaching conclusions

viii kind. Our tools, of course, are the materials'pters II and 24.

tilesOnc major emphasis is that statistical conclu-

*tied on samples, are probabilistic in na-Alert is enough material developed in the

so that the student should be able toaid the meaning of statistical statements'WAR encounter in the future. In order tosuch understanding, the student shouldto follow the general arguments of theThe binomial case is emphasized, since

45

this provides the simplest example of a situatiin which a hypothesis may be tested by sampliAfter finishing the chapter, the student shouldable to state a simple hypothesis and to andstand that experimental evidence could leadrejection or acceptance of the hypothesis. Mastudents may eventually forget the detailsexample, that as = Vnpq for a binomial distrition), but the experiences should enable themlater years to exercise intelligent judgmentevaluating assertions based on statistical eviden

In particular, he will have sufficient backgrouto move later into a more traditional courseintroductory statistics and probability givenhaps at the senior high school or college freshlevel. Another possibility is for the interested sdent to study the programmed course givenintroduction to Probability, School MathemaStudy Group, Stanford University, 1965. Part Ithis text offers a careful treatment of independevents, based on the important notion of contianal probability.

General CommentsI. Since the curriculum is for "alI" students,

objectives of exercising, strengthening, andviewing previously learned materials need tokept in mind.

2. The objectives of estimating and approximing have not been mentioned.

3. A number of the chapter statements omit mtioning the skill objectives which are built ithe text.

4. An overall objective is extending from omathematical system to a larger sone.

46

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