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A SURVEY OF THE MATHEMATICS CURRICULUM T IN THE MONTANA SECONDARY SCHOOLS A Thesis Presented to the Faculty of Western Montana College of Education In Partial Fulfillment of the Requirements for the Degree Master of Science in Education by J. Gordon Donovan September 1963 APPROVAL Advisor (. ACKNOWLEDGEMENTS The writer wishes to acknowledge the help of the following teachers; for without their assistance this study would have been impossible. Contributing teaohers Robert Graham Andrew McDermott Raymond Shackleford Glenn Thomas Darrell McCracken l1arvin Hash Gary Boyles Louis Stahl Gerald Downing Noel Teegarden Darle Hemmy James Tryon Ruth Ulmen Donald Hilla Karl Fiske Paul Aspevig Gary Evans Florence Timmerman James Muck Richard Walker Allan Hopper Sharon Robertson Edward Goodan Vernon Herbel LaVerne Frantzich Lee Von Kuster Elsie McGarvey Darrell Meskimen Sr. Marie Ferring Rev. William Allen Joseph Wolpert Cecilia Klofstad Vernon Pacovsky Donald Owen Carl Fox Sheila Wiley James McCulloh Glenn Pearson Howard McCrea Bernard MacDonald William Sweet Arlen Se~an Louis Karhi William Conners Harold Contway Arthur Baumann Kathleen Holm William Ross Joseph Cullen Dennis Olin Paul Ornberg Stanley Rasmussen Allan Skillman Thomas O'Neil Joseph Israel Ronald Steffani Paul Arneson Ronald Soiseth Phyllis Washburn Marjorie Von Bergen John Shular vIilliam Chalmers Claude Foster Harold Selvig John Oberlitner Douglas Vagg George Scott Lester Paro Edwin Goyette Donald Cole Doyle Coats Jerome Knopik Arline Hofland James Wood Delmar Klundt Walter Scott Ed! th Miller Nina Roatch Harry Baker Theodore Bergum Russell Hartford Mildred Schow Warner Fellbaum Leonard Amundson Sr. John Berchmans Leist Sr. M. Griswalda Norman Cascaden Sr. Judith Ann TABLE OF CONTENTS CHAPTER I. THE PROBLEM AND DEFINITIONS OF TIRMS USED • • • • • • PAGE 1 2 2 3 S 5 6 6 6 6 6 6 II. The Problem •••••••••• Statement of the problem ••• • • • • • • • • • • • • • • • • • • Importance of the study • Definitions of Terms Used • • • • • • • • • • • • • • • • • • • • • • • • • Secondary schools • • • • • • • • • • • • • • • • Mathematics curriculum • • • • • • • Mathematical topics • • • • • • • • • • • • • Teacher opinions •••••••••••• • • • • • • • • • • • • Senior mathematics • • • • • • • • • • • • • • • Fundamental operations Math club • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • Basic Assumptions and Limitations • • • • • • • • • 6 Assumptions • • • • • • • • • • • • • Limitations ••••••••••••• Sunur!B.ry • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • REVIEW OF THE LI TFRA TUItE • • • • • • • • • • • • • • A Comparison of the Two Philosophies of Mathematics Education • • • • • • • • • • • • • • 6 7 7 8 8 Progrants • • • • • • • • • • • • • • • • • • • • • 9 Literature on summer mathematics programs for the mathematically talented • • • • • • • • 10 CHAPTER Instructional program • • • • • • • • • • • • • • Evaluation of the program • • • • • • • • • • • • Experimental program in mathematics • • • • • • • • Prerequisites for the program • • • • • • • • • • What were some of the advantages ••• Pro blems • • • • • • • ~' • • • • • • • Modern mathematics in the senior year Problems Study Groups • • • ••• • • • • • • • • • • • • • • • • • • • • • • • • • The School Mathematics Study Group •• Uni versi ty of Illinois Curriculum Study • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • in Mathematics ••••••••• • • • • • • • University of Mar,yland Mathematics Project ••• Boston College Mathematics Institute • • • • • • • • • • Commission on Mathematics of the College Entrance Examination Board • • • • • • Developmental project in secondary mathematics Southern Illinois Universit,y The Secondary School Curriculum Committee National Council ot Teachers ot Mathematics • • • • • Sllmmary' • • • • • • • • • • • • • • • • • • • • • v PAGE 11 14 15 15 17 18 18 21 24 24 24 24 2$ 2$ 2$ 2$ 26 CHAPTER III. PROCEDURE AND METHOD OF RESEARCH • • • • • • • • • • Procedure • • • • • ••• • • • • • • • • • • • • • Questionnaire inquiries • • • • • • • • • • • • • Groups surveyed • • • • • • • • • • • • • • • • • Editing the inquiries • • • • • • • • • • • • • • Compiled data • • • • • • • • • • • • • • • ••• Analysis of the problem SubProblems of the Study • • • • • • • • • • • • • • • •• • • • • • • • • • Subproblem A • • • • • • • • • • • • • • • • 0 • Data necessary ••• • • • • • • • • • • • • • Sources of data. • • • • • • • • • • • • • • • Analysis of data Subproblem B • • Data necessary Source of data Analysis of data • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • Subproblem C • • • • • • • • • • • • • • • • • • Data necessary • • • • • • • • • • • • • • • • Sources of data • • • • • • • • • • • • • • • • Analysis of data •••••••••••••• • Subproblem D • • • • • • • • • • • • • • • • • • Data necessary • • • • • • • • • • • • • • • • Sources of data • • • • • • • • • • • ••••• Analysis of data • • • • • • • • • • • • ••• vi PAGE 28 28 28 28 )0 30 31 32 32 32 33 33 34 34 3, 35 36 36 36 36 36 37 37 37 CHAPTER IV. v. SllJll11l8.r:Y' • • • • • • • • • • • • • • • • • • • • • • THE GENERAL STRUCTURE OF THE fiLA THEMATICS . CURRICULUM • • • • • • • • • • • • • • • • • • • • The Percentage of Students Enrolled in Mathematics Classes • • • • • • • • • • • • • • • Breakdown of curriculum • • • • • • • • • • • • • • General mathematics • Algebra • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • Geometry Algebra II • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • Trigonametr,y and solid geometry • • • • • • • • • • Senior mathematics • • • • • • • • • • • • • • • • Mathematics for the noncollege capable • • • • • • Textbook Criteria • • • • • •••••••••••• Algebra and algebra II • • • • • • • • • • • • • Geometry • • • • • • • • • • • • • • • • • • • • Textbooks used by Montana aeconda~J schools • • • Summary •••••• • • • • • • • • • • • • • • • • School enrollment in mathematics classes • • • • Mathematics for the noncollege capable • • • • • Textbooks used • • • • • • • • • • • • • • • • • ~ATHEMATICAL TOPICS • • • • • • • • • • • • • •••• vii PAGE 31 40 40 40 41 44 44 46 46 49 53 " 55 56 57 60 68 68 69 11 CHAPTER viii PAG~~ Background • • • • • • • • • • • • • • • • • • • • 71 Mathematical vocabulary • • • • • • • • • • • • • • Signed numbers •••••••••••••••••• Fractions • • • • • • • • • • • • • • • • • • • • • Geometric concepts Solving equations • • • • • • • Evaluation formulas • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • 72 74~ 75 76 71 78 Verbal problems • • • • • • • • • • • • • • • • • • 79 Graphing • • • • • • • • • • • • Solving systems of equations' •• Fundamental operations ••••• • • • • • • • • • • • • • • • • • • • • • • • • • • • Sets • • • • • • • • • • • • • • • • • • • • • • • Serles • • • • • • • • • • • • • • • • • • • • • • 80 81 81 83 84 Neatness of papers ••• • • • • • • • • • • • •• 84 Checking work • • • • • • • • • • • • • • • • • • • 85 Sununary • • • • • • • • • • • • • • • • • • • • • • VI. TEACHER OPINION ON THE MATP~:MATICS CURRICULUM • • • • 86 88 88 89 95 VII. Background • • • • • • • • • • • • • • • • • • • • Teacher opinions on curriculum ••• • • • • Opinions on the elementary school program • • • • • • • • Sunnnary • • • • • • • • • • • • • • • • • • • • •• 98 TEACHER ORIENTATION • • •••• • • • • • • • • • •• 100 CHAPTF..R ix PAGE Background • • • • • • • • • • • • • • • • • • •• 100 Teacher opinion • • • • • • • • • • • • • • • • •• 101 An inservice course for elementary teachers • • • MATHEMATICS FOR ELEMENTARY TEACHERS • • • • • • • • • 101 10) Assignment I •••••••••• • • • • • • • •• 106 Problems • • • • • • • • • • • • • • • • • • •• 107 Vooabulary • • • • • • • • • • • • • • • • • • • • Nurnber . . . . . . . . . . . . . . . . . . . . ~ Numeral • • • • • • • • • • • • • • • • • • • • • Natural nurabers • • • • • • • • • • • • _ • • Q 0 Whole numbers • • • • • • • • • • • • • 108 108 108 108 108 Nonnega ti ve rational nwnbers • • • • • '. • • •• 109 As signment II • • • • • • • • •.• • • • • • • • ... 109 Problems • • • • • • • • • • • • • • • • • • • • Assignment III  • • • • • • • • •• • • • • • • • • Problems Set I' • • • • • • • • • • • • • 0 • • • Problems Set II • • • • • • • • • • • • • • • • • Problems Set n! • • • • • • • • • • • • • • • • Assignment IV • • • • • • • • • • • • • • • • • • The number one • • • • • • • • • • • • • • • • • 109 110 III 112 113 114 114 Properties of the number, one ••• • • • • • •• 114 Different names for one • • • • • • • • • • • • • The number zero • • • • • • • • • • • • • • • • • 11, llS CHAPTER Properties of the number zero • • • • • • • • • • Problems •• • • • • • • • • • • • • • • • • • • Assignment V Definitions • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • Point Space Linea Plane · . . . . ~ . . ~ . . . . . . . . . . . . • • • • • • • • • • • • • • e • • • • • • e • • • • • • • e • • • • • • • • • • • • • • • • e • • • ., ., • • eo ., Skew lines • • • • • • • • • • • • • • • • • • x PAGE 115 115 116 117 117 117 117 111 117 Ray • • • e • • • • • • • • • • • • • • • • •• 117 Angle • • • • • ~ • • ~ ~ • • • • • e • • • •• 111 Vertex Triangle Class work • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • It • • • • • • • • • • • • • • • • 111 111 117 Points to pond~r • • e • • • • • • • • • • • • ~ 119 SUlllID3.ry" • • • • • • • • • • • • • • • • • • • • •• 119 VIII. Sm~RYJ CONCLUSIONS, AND RECO~1E~~ATIONS • • e • •• 121 Summar.y and Conclusions • • • • e • • • • • • • •• 121 Need ••••••••• • • • • • • • • • • • •• 121 Problem • • Assumptions Limitations · . . . . . . . . . . . . . . . ~ . . • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • Qeneral structure of the mathematics curriculu.~ • 123 123 123 124 CHAPTER Mathematics for the noncollege Xi PAGE capability • • • • • • • • • • • • • • • • •• 12, Textbooks ••• • • • • • • • • • • • • • • • • • 126 Mathematical topios • • • • • • • • • • • • • •• 127 Teacher opinion • • • • • ~ RECOMI·1ENDATIONS ••••••• • • • • • • . . . . .. • • • • • • • • • • • Mathematics program • • • • • • • • • • • • • • • Review or 'curriculum • • • • • • • • • • • • • • Support of school author! ties • Review of starr'assignments •• • • • • • • • . •. . . . . . • • • • Program for teacher improvement • • • • • • • • • Money for mathematics education • • • . ... . BI BLIOGRAPHY •••• • • • • • • • • • • • • • APPENDIX A, LETTERS '. • • • • • • • • • • • • • APPENDIX B, QUESTIONNAIRE • • • • • • • • • • • · .• · '. • • • • 128 130 131 133 133 133 134 134 131 141 144 LIST OF TABLES TABLE PAGE I. Breakdown of Advanced Mathematics Programs in Montana Secondary Schools • • • • • • • • • • • • ~ 52 II. ,Teacher Opinion on when General or Consumer Mathematics should be Taught in the Schools • • • • •. • .. • • • • • • • • • • • • • • • ,0 III. Distribution of Schools with regard to Frequency of Ordering New Mathematics Textbooks • • • • • • • • • • • • • • • • • • • • • • IV. Breakdown of Schools Ordering New Algebra and Algebra II Textbooks during the Period . . ~ ~ . 1963 to 1967 • • • • • • • • • • • • • • • • • • • • V. Degree of Difficulty of Mathematical Topics' for Students to Handle· as Indicated by Teachers of Mathematics • • • • • • • • • • • • • Ct • VI. Schools having Orientation Programs for 54 58 59 73 Elementary Teachers • • • • • • • • • • • • • • • • 0 102 LIST OF FIGURES namm 1. Average Percentage of Total School Enrollment Attending Mathematics Classes •••• ~ • • • • • • • • • • • • • 2. Number of Schools Showing Breakdown ot Total School Enrollment ,Attending General or • • • • • Consumer Mathematics Classes • • • • • • • • • • • • • 3. Number of Schools Showing Breakdown of Total School Enrollment Attending Algebra Class • • • • • • 4. Number of Schools Showing Breakdown of Total School Enrollment Attending Geometry Class • ,. Number of Schools Showing Breakdown of Total 6. School Enrollment Attending Algebra II Class • • • • • • • • • • • • • • • • • • • Number of Schools Showing Breakdown of Total School Enrollment Attending Trigonometr,y and Solid Geometry Classes • • • • • • • • • 1. Number or Schools Showing Breakdown of To~l Sohool Enrollment Attending Senior • • • • • • • • • • • • • • • Mathematics Class • • • • • • • • • • • • • • • • • • 8. Textbooks used by Schools Offering General and/or Consumer Mathematics • • • • • • • • • • • • • PAGE 42 43 45 41 48 50 ,1 61 xiv FIGURE PAGE 9. Textbooks Used by Schools Offering Algebra • • • • • • • • • • • • • • • • • • • • • • • 62 10. Textbooks Used by Schools Offering Plain Geometry ••••• • • • • • • • • • • • • • • • 11. Textbooks Used by Schools Offering Algebra II • • • • • • • • • • • • • • • • • • • • • • 64 12. Textbooks Used by Schools Offering Trigonometr.y and Solid Geometry • • • • • • • • • • • 65 13. Textbooks Used by Schools Offering Senior Mathematics • • •• • • • • • • • • • • • • • • 66 CHAPTER I THE PROBLE'M AND DEFINITIONS OF TERMS USED Cultural changes occurring in present time have had significant implications for the curriculum as a whole and for mathematics in particular. Before the trend toward abstraction was felt, mathematics in the public school was cheitly concerned with manipulative operations and the skill acquired in performing the operations within a mathematical system rather than with an understanding of the properties of the system. The difference in point of view between the older approaoh and the modern conception was well put by Sawyer: The mathematician of older times asked, "Can I find a trick to solve this problem?" It he could not find a trick today, he looked for one tomorrow. But ••• we no longer assume that a trick need exist at all. We ask rather, "Is there any reaeon to suppose that this problem can be solved with the means we have at hand? Can it be broken up into Simpler problems? What is it that makes the problem soluble, and how can we test for aolubili ty?1t We try to discover the nature of the problem we are dealing with.l The new developments in graduate and research mathematics implied a necessary shift in emphasis for secondar.y school mathematics. The new applications of mathematics signified new problem lw. W. Sawyer, Prelude to Mathematics (Baltimore: Pelican Books, Penguin Books, Inc., 19~), p. 214. 2 material. Probability, statistical inference, finite mathematical structure, linear programwingall indicated a change in the applications and useful purposes of mathematics. The changes in the cultural, industrial, and economic patterns of many nations called for a change in educational patterns. More people should have been better trained in scientific knowledge. Even the coumon layman must have the ability to understand science in the world of today and knowing mathematics is basic to understanding science. In light of the great development in mathematics over the past thirty yea~sJ we should have examined this subject critically and carefully. The changes which those developments signify have failed to be adequately reflected ~n the present secondary school curriculum. THE PROBLEM Statement 2!..!ill! problem. It was the purpose of this study to examine the mathematics curriculum in the secondary schools of Montana and to see what improvements could be made to adjust to the new developments in mathematics. Four areas were considered, What is the general structure of the mathematics curriculum in the Montana secondary schools? What mathematical concepts are the most difficult for the student to master? r~t are the opinions of teachers toward the mathematics 3 curriculum? What is being done to orientate elementary teaohers on new mathematical concepts. Importance 2!~ stu91 The unprecedented growth of pure and applied mathematics in the United States has caused an acute shortage of good mathematicians. Supplying this demand has become a knotty problem. Mathematicians need more training than ever before. Yet, they cannot afford to spend more years in school, for mathematicians are generally most creative when very young.2 A whole new concept of mathematical education, starting as early as the third grade, has appeared as the most logical way to solve this problem. Never before have so many people applied such abstract mathematics to so great a variety of problems. To meet the demands of industry, technology, and other sciences, mathematicians have had to invent new branches of mathematics and expand old ones. They have developed new statistical methods tor controlling quality in highspeed industrial mass production. They have created an elaborate theor.y of information that enables communications engineers to evaluate precisely telephone, radio and television circuits. They have analyzed the design of automatio controls for such complicated systems as factory production lines and supersonic aircraft. 2 George A. Boehn and others, The New World of Math (New York: The Dial Press, 1959), p:51:   4 Faced with the problems of "instructing" computers what to do and how to do it, mathematicians have reopened an old and part~ dormant field} Boolean algebra. This branch of mathematics reduced the rules of logic to algebraic form. Numerical analysis, a main part of the stu~y of approximations, is another field for computers. Wi th mathematics expanding the way it has in the past few years, the need to revise the mathematics curriculum in the secondary schools has been made apparent. Young people in high sohool do not always know what careers theY' will follow. v1hat they should know beyond all doubt 1s that lack of mathematical preparation has closed man.y doorsnot only doors that open the way to engineering and the natural sciences, but also newer doors that lead to important areas of the social sciences, biological sciences, business, and industry.) Mathematics has come easier to the young. The study of some subjects may have been postponed ~dthout serious lOBS; age could have made their appreciation much 'easier. Mathematics has been differentfor the young it has been now or never. Experience has proven that high school was the place where most of our great scientists and mathematicians first acquired the interest that 3Ibid., pp. 6315. spurred them on to high aohievement. 4 Ever.y child should at least be given the opportunity to learn the mathematical concepts and skills which will make these activities accessible to him. Our task has been to provide every pupil today with the ~athematical instruction which will be most beneficial to him tomorrow and make him most useful to society. In view ot this fact and the urgency of the Situation, it becomes imperative that Montana administrators and school officials have a knowledge of the mathematics structure in the secondary schools and how mathematics teachers have reacted to the new material and concepts that have been introduced. It was hoped that this study would provide some insight into the mathematical structure of the secondar.y schools in Montana. DEFINITIONS OF TERNS USED Seconda;r schools. Public bigh schools, county high schools and private schools as designated in the Director" of Science and I~thematics Teachers of Montana, 196263; NDEAIII70~1/62,O (Revised) from the State Department of Public Instruction. 4Commission on Mathematics, Program for College preparato~ Mathematics (New York: College Entrance Examination Board, 1959, p. 12. 6 Mathematics curriculum. The mathematics courses offered in the secondar,y schools, including mathematics clubs, and extracurricular acti vi. ties dealing vIi th the subject of mathematics. Mathematical topics. This refers to the list of topics under question 1$ of the questionnaire. (See appendix). Teachers opinions. Those corr~ents which were noted by teachers in answer to question 20 of the questionnaire. (See appendix). Senior Mathematics. Advanced mathematics courses, other than trigonometry and solid geometr,y, offered in the 12th grade. Fundamental operations. The operations of addition, subtraction, multiplication and division. Math club. A student organization with the purpose to create and maintain interes~ in mathematics. BASIC ASSm1PTIONS AND LIHITATIONS Assumptions. It was assumed that the mathematical curriculum was designed for pupils of all ability levels since all students would live in the same technological culture. It was to be expected that different students would attain different levels of understanding and skill. It was assumed that the curriculum should provide materials and experiences which take 1 into account these different levels. Limitations. This study was limited to an overview of the mathematical structure in the seeondar,y schools rather than to try to deal with the multitude of different problems which would confront each individual "school. The wri tar indicated that teacher response would be best sui ted for this study. The study was limited to the one hundred ninety public, county, and private schools as designated by the director" of Science and Mathematios Teachers of Montana, 196263NDEAIII7011/6250 (Revised). Summary. The new mathematics has played a major role in the development of the complex technological world that we know today. Unfortunately for the nation's youth, this prodigious advance in mathematics education. The implementation of a revision of the mathematics curriculum has been taking place with unprecedented speed. It was the purpose of this study to examine the mathematics ourriculum in the secondary schools or Montana through means of a questionnaire on the following areas(l) general structure, (2) mathematical concepts, (3) teacher opinion, and (u) teacher orientation.to determine the degree of mathematical revision in the curriculum. CHAPTER II REVIEW OF THE LITERATURE It has been suggested that a person should study the historical development ot mathematics trom its rudimentar.y beginnings to the modern abstract form in order to gain understanding and cultural appreciation of the subject. In 1940 two separate investigating committees published reports on the ph11050PQY of mathematics. On the surface, these reports seem quite divergent. However, recent studies have confirmed that a convergence of both these philosophical surveys gives promiB.e of the best curriculum. The following chart summarizes these findings.1 A COMPARISON OF THE TWO PHILOSOPHIES OF MATHEMATICS EDUCATION: Hathematics in General Education Child Society Universe DICHOTOMY OR CONVERGENCE ORDER OF IMPORTANCE The Place of Mathematics ~ SecondarjEducation PQysical Universe Society Child lHoward F. Febr, Teaching High School Mathematics (Washington, National Education Association, 1955), p. 8. Personal living Personalsooial living Socialc1 vic rela tiona EConomiocareer relationships which call for , OBJECTIVES So01al sensitivityeethetic8 Toleraneecooperativeness Selfdirectioncreativenees Reflective thinking Formulation and solution Data Approximation Function Proof Symbolism Operation Changing values and SUBJECT MATTER PRIMARY VALUES Problemsolving ability Field p5,1chologyanalysis insight A philosophicconfigurational integrated learning program Difficul t to evalua te PRINCIPAL LEARNING ASPF:CTS TFACHING D1PLICATIONS PROGRAMS 9 Abili ty to think olearly Information, concepts, principles FUndamental skills Attitudes I nterests and appreciations Field of number Geometric form, space perception Graphio Representation Elementary analysis Logical thinking Relational thinking Symbolic representation Permanent values and Organized subject matter Association and generalization A wellstructured and organized subjectmatter program Easily evaluated Much has been written in regard to the uses or modern mathematics, the significanoe of modern mathematios for the secondary school curriculum, and the acute shortage of mathematics 10 teachers and mathematicians, both pure and applied. While much has been said about the various needs in mathematical fields, there has been very little done in the way or experimental mathematics programs tor secondary students to learn the concepts of modern mathematics. The number of programs that have been carried out are few and seem to fall into three oategories I Summer institutes, the teaching ot algebra in the eighth grade so that modern concepts could be taught during the higb school years, and programs'for twelfth grade students only. A brief summary of a progra.m in each one of these categories will be given. Literature S!!! Summer Mathematics Programs ~~ Mathematically Talented A number of summer institutes for the mathematically talented have been carried out across the nation. As all of them have been quite 8i~ilar in format and purpose, only the summer program at Florida State University, Tallahassee, Florida was outlined. The program was six weeks in duration and met for classes four to five hours daily. , The purpose of the program wass 1. To identify high school youngsters capable of becoming research mathematicians or exceptional mathematics teaohers. 2. To enhance and develop the interests of these young men and women by providing them with new insights into the expanding field of mathematical knowledge. n 3. To bring the youngsters into contact with mathematical knowledge of a kind not found in conventional high school courses, thus providing the students with opportunities to engage in creative mathematical activities. The students were divided into two groups. Group I was composed of students who had completed no more than the ninth grade by June 1959, and who had finished at least one year of algebra. Group II was composed of students who had completed the eleventh grade by June 1959 and had finished at least three years of college preparatory mathematics. Instructional Program. I. Programming the 1m! 650  3 weeks A. Introduotion 1. The memory un! t 2. The arithmetic unit 3., The control unit 4. Instruction format B. Coding 1. Arithmetical operations 2. Shirt operations 3. Branch operations 4. Watson System special operations c. Programming 1. Use of Loops 2. Setting initial condition .3. Terminal Loops 4. Speed verS\1S space D. Precision, scaling, and testing 1. Notation for fixed point scaling 2. Floating point calculations 3. Double precision calculations 4. Tracing II. Numerical Methods A. Newton's i tera ti ve method 1. Coordinate geometr.y 2. . Slope or the tangent line to a curve 3. Cube root routines B. Approximation ot areas by finite sums (Skipped for Group I) III. Problems in Programming A. Compute Phi B. Solve systems of linear equations in ten unknowns c. Compute cube roots of numbers from 1 to 10,000 to ten significant digits IV. Number theoryGroup I14 weeke. Test: Theory 2! NUmbers by Burton Jones. A. Group II 1. Integers 2. The Euclidean Algorithm 3. Greatest common divisor 4. Unique factorization into primes S. Diophantine equations 6. Congruences 1. Elementar,y properties and theorem 8. Chinese remainder theorem 9. Order 10. Fermat and Euler theorems 11. Primitive elements 12. Properties of Euler and Phi Function 13. Field axioms 114,.. Polynomials F(x) and unique factorization into irreducible polynomials 16. La Grange's theorem 12 17. Primitive roots mod 2 as applications 18. Congruence in F(x) and the proof that F(x) mod an irreducible polyno~~al is a field 19. The concepts of prime fields, extension fields, and root fields 20. The construction of projective geometries from fields 21. Elementary aspects of groups or sets 22. La Qrange~s theorem B. Group I  2 weeks 1. Integers 2. Euclidean Algorithm 3. Unique factorization into primes 4. Diophantine equations 5. Congruences 6. Chinese remainder theorem 7. Order 8. Reduced residue systems 9. Euler's theorem 10. Properties ot EUler's Phi Function 13 V. Probability and Statistics  6 weeks  Texta Introductory Probability ~ Statistical Inference~ Experimental Course. A. Organization 1. Presentation of data 2. Frequency distributions B. Summarizing a set or measurements 1. Mean of the distribution 2. Standard deviation of the distribution c. An intuitive approach to probability D. Formal approach to probability E. The laws of chance for repeated trials F. Applications of the binomial distribution and acceptance sampling and testing hypotheses G. Using samples for estimation and sampling from fim te popua tiona H. Mathel'r.a tical induction VI. Algebra  Group I and Group II  4 weeks A. Definition of a field B. Concept of a set; one to one correspondence o. Distinction between number and numeral D. Theoretic study of number systems 1. Natural numbers 2. Integers ). Rationals 4. Irrationals 5. Complex numbers E. Matrices 1. Translations, rotations, and stretohes 2. Definition and properties ot matrix mul tiplication and addition 14 3.  Solution of linear equation systems via matricies F. Symbolic logic and functions and relations 1. Statements and connective 2. Sentences and quantifers 3. Sets 4., Functions S. Relations 6. Functions and operations VII. Introduction to Russian  Group I and Group II  6 weeks  Testa Simplified Russian Grammer by Fayer, Pressman, and Pressman  )8 lessons covered. Evaluation ~ ~ program. All the objectives that were set at the beginning or the program were obtained to a high degree. The students took the STEP Mathematics Test, College Level, at the end of the summer program. The raw scores made by the students ranged from 36 to 50, and the median raw score for the group was 43 out of a maximum of 50 possible. On the College Qualification Tests, Combined Booklet Edition, the range was between 2, and 99 percent with 93 the median percentile rank for the group. 15 Group I took the Wechsler Intelligence Scale for Children and Group II took the Weohsler Adult Intelligence Scale. The range in deviation IQ' 5 was 122 to 152, with the mean IQ of the group at 132.2 Experinlent~ ProgrllIl! !!! Mathematics The Tulsa Publio Schools in Tulsa, Oklahoma started an experimental mathematios program in the spring of 1933. Two tests were given to seventh grade students near the end of that sohool year to determine placement in the experimental program. Prerequisites ~ !h! program. 1. Student must rank at the ninth grade level or above on the Arithmetic Reasoning section of the Stanford Achievement Test. 2. Student must rank about the 90th percentile on the California Algebra Aptitude Test. 3. Student must be interested in mathematics and plan to major in mathematics in high school. 2Eugene D. Nichols; "A Suw~er Mathematics Program for the Mathematically Talented, tt .!!!! Mathematics Teacher" LIll, (April, 1960), 23,. 4. Student must have written approval of the parents. 5. Student must have a reoommendation from his seventh grade teacher based on the student's IQ, work habits, and his enthusiasm for mathematical work. The average grade level of the group on the Stanford Achievement Test was 10.1 and the average on the california Algebra Aptitude Test was 93.). The average IQ was 119.0. The first three weeks of the course was devoted to an intensive study of eighth grade work. The remainder of the year was spent on traditional work in freshman algebra. The Stanford Acheivement Test, the Lankton First Year Algebra Test, and the Cooperative First Year Algebra Test were given in the spring or 1956. Average scores of the group were a8 follows: Stanlord Achievement Test 12.0 Lankton First Year Algebra Test 92.3 percent Cooperative First Year Algebra Teet 95.5 percent This was considerably higher than scores made by the regular ninth grade students. In the ninth grade, the students were given a review of basic algebra, linear equations in 1, 2 and 3 unknowns, factoring, rra~t1onal and negative exponents, quadratic equations, determinente, logarithms, imaginary and complex numbers, binomial theorem, progressions and series, powers and roots, mathematical 17 induction, variations, theory of equations, graphs of functions in 1, 2, and 3 unknowns, and probability. The average percentile rank achieved by the students on the Cooperative Advance Algebra Test, given in the spring of 1956 was 92.1. As sophomores, the group took 3/5 of a year of plane geometry and 2/, of a year of solid geometry. Their average score on the test given at the end of the session on plane geometry was 93.6 and the average score of the solid geometry test was 94.8. During their junior year, the group took trigonometry and college algebra, using a textbook in which the subjects were more or less integrated. They obtain~d an average rank of 79.6 on the Cooperative Trigonometry Test. Mathematical analysis, with some attention to probability and statistical inferenoe, was studied during the senior year. Of the original 60 pupils that started the program, 35.remained. These students consistently outscored students who had not been exposed to the advance program. Of the 24 students who dropped out, 15 had transferred to a different school. ~ ~ ~ £!. ~ advantages? 1. The students were enthusiastic about mathematics and had the necessary push to do the work required. 2. The students were capable. All the tests given indicated that they did better work than regular students 18 who were one year older. 3. They received an additional year's work in mathematios. This provided them with an adequate background tor almost any college course. 4. Because of their mathematics background, they were able to do better work in physical science courses. Problems. 1. Some of the students were not sufficiently matured and became emotionally disturbed at trivial upsets or when something went wrong. 2. There was a tendency for some of the students to feel that they belonged to a superior group. Snobbishness developed. 3. The teachers had a tendenoy to expect too much from the group in later courses. 4. Parents sometimes pressured their child beyond his capabilities. Emotional disturbances resulted.3 Modern Mathematics in the Senior Year .......      The following program was used in the twelfth grade at Wisconsin High School, Madison, Wisconsin in 19$6 and 1957. 3Coy C. Pruitt .. nAn Experimental Program in Mathematics," The Mathematics Teacher, LIll, (February, 1960), 102. 19 The text used was Principles £! Mathematics by Alendoerfer and Oakly, published by HcGrawHill Company. This text was followed closely during the entire course. While the teacher was a bit apprehensive, the students were not afraid at all and liked the idea of learning something altogether different from anything they had seen before. Logic and the number system were the first subjects tackled, and though logic and the number system are not "modern", these topics were presented from a "modern" point of view. This manner of presentation gave the students the tools and the view point necessary to study groups, fields, sets, and Boolian algebra. The students not only learned ne~ ideas but they also shook themselves free from the rigid, traditional approach to traditional algebra. They began to understand and appreciate the nature of mathematics through this new freedom. Chapter VI in the text dealt with functions and while the students had been exposed to functions in elementary algebra, it was here that they learned what function meant. This unit set the pattern and laid the foundation for the more tradi tional work that followed. The next three chapters were on algebraic funotions, trigonometric functions, and exponential and logarithmic functions. These chapters taught the students to use algebra, if nothing else, and gave the teacher a breather before proceeding into analytic 20 geometry, limits and calculus. The chapter on limits ranked along with logic and functions as the most important chapters in the course. The concept of limi t seemed to be the most difficult one for the students to grasp. Yet, they finally mastered it because they worked the hardest on that portion of material. The last chapter in the book dealt with statistics and probability. This chapter, dealing with descriptive statistics, was the most enjoyable one in the book because of its simple arithematic calculations and easily grasped conoepts. The teacher and students were fully aware at all times that their purpose was not to learn all about groups, sets, limits, statistics, and logic. The concept of taking a look at the broad field of mathematics, examining some parts in detail, and trying to learn the relationship of courses, both past and future, was fully realized. Naturally it can never be proven that a different course, traditional or otherwise, would have been of greater value. These students llere sophomores in college in 1959. Some were excused from taking any college mathematics because they scored very high on placement tests. Others, who chose mathematics for their college 'Work, began on an advanced basis. All but one of these made A'S in mathematics the first semester. They have continued to maintain high standings in college mathematics 21 courses. Some students have said that their nightly struggles with mathematics in the twelfth grade course have proved invaluable in terms of content and study habits. The teacher believes that the text used in this course could be ueed in any twelfth grade mathematics class. It would be necessar,r to omit some material and emphasize different parte of the text, depending upon the pupil's background and needs. The following problems, taken from semester examinations given during the course, show the content of the course and the side coverage of mathematical material. Problems. 1. \~ri te in "if then form": A sufficient condition that Lake Huron freeze is that the temperature be below 0 degrees F. 2. If the light is red, the ear will stop. The car does not stop. ~JNCLUSIONI The light is not red. Is this a valid conclusion? 3. !t'ind an integer x such that: x + 1 • 4 (mod 3) o~ X .c::J and 4. What real values of x satisfy 6x2 + 5x  4 :::::,01 5. Show by a series of Venn diagramst (A' fl B) I AU B' 6. Graph: 7. State the Fundamental Theorem of Algebra. 8. The inverse of the converse is the • ·9. Are identical sets equivalent? 10. In Boolean algebra (0, 1), does (1 O)X 1 11 11. rex) and g(x) are functions of x. What is g(f) called? 12. Describe the steps used in finding 53•2 b.Y logs. 13. Wri te the equa. tion for the line through (2,3) which is perpendicular to the line through (7,2) and (1,4). 14. Sn = 3  ~ is the general term of a ,sequence. Guess tne limit of the sequence and prove that this is the limit. 15. Find the first and second derivatives of rex): rex) • 8~ + j  3. 16. Find SS (3u2 + 2u)du 1 11. Calculate: mean, median, and standard deviation for this set: 5,3,3,1.9,4,6,1,2,5. 18. Prove that rex) is or is not continuous a x = ls r(x) .. ~i + x 2 , 2, X a 0 x/:o 19. vlhat is the probability of getting at least two 6t s in 2 rolls of 3 dice? 20. Prove by mathematical induction: am 4 a + ar + ar2 + • • • + arn • a  • • lr 4Joe Kennedy, "liodern Mathematics in the 12th Grade," ~ }~thematics Teacher, LII, (Februar,y, 1959), 97. 22 23 The new emphasis in the study of algebra was upon the understanding of the fundamental ideas and concepts of the subject such as the nature of number systems, ~~ctionsin particular, the linear, quadratic, exponential and logarithmic, and identities and inequalities. One way to foster an emphasis upon understanding and meaning was through the introduction of deductive reasoning. Not all reasoning is syllogistic or deductive reasoning. Training in mathematics based on deductive logic does not necessarily lead to an increased ability to argue logically in situations where insufficient data e}asted. Deductive methods are taught prima~ily to enable the pupil to learn mathematics. The subject matter contained in the courses presented was of comparatively recent development in mathematics as far as presentation of material was concerned. In the early 1940's, the study of groups and fields filtered down into the upper undergraduate years from the graduate school, and more recently, into the lower mathematios courses. Carefully seleoted material of this sort was found to be within the grasp of able bigh school students. Experience has indicated that these students have found t~~s subject matter both challenging and interesting. A course in modern mathematics would serve as an admirable means by whioh to bridge the gap between high school and college mathematics. 24 Study Groups !h! School Mathematics Study Group. This group represents the 1argest united effort for improvement in the history of mathematics education. During the school year 195960, sample textbooks and teachers' manuals for grades seven through twelve were tried out in fortyfive states by more than 400 teachers and 1,2,000 pupils. The SMSG textbooks contain new topics as well as changes in the organization and presentation of older topics. Attention was focused on important matt~mat1cal facts and skills and on basic principles that provide a logical framework for them. University of Illinois Curriculum Study ~ Mathematics. Work on the UICSM ~Aterial began in 1952 and by the 195960 school year had been used experimentally in twentyfive states by 200 teachers and 10,000 pupils. The textbooks emphasized oonsistency, precision of language, structure of matheu~tios, and understanding of basic principles through student discovery. University 2!. Maryland Mathemati.cs Projeot. The program was used in ten states by 100 teachers and 5,000 students. The courses were designed to bridge the gap between arithmetic and high school mathematics. Some of the chapter titles of the seventh grade textbook weret "Symbols", "One and Zero", 25 "Mathematical Systems", "Logic and Number Sentences."5 Boston College Mathematics Institute. Historical development is used to break away from the traditional approach and also to give the student an opportuni ty to use his Olm initiative and creativity. Mathematics was studied through problems that confronted primi ti ve If!an up through present day questions that confront mathematicians. Co~~s5ion ~MathematicB of ~ College Entrance Examination Board. The commission made recommendations looking toward the mod.ernization, modification) and improvement of the oollege preparatory mathematics curriculum in the secondary schools. The objective was to produce a curriculum suitable for students and oriented to the needs of mathematics, science, business, and industry in the second half of the 20th century. Development Project ~ Secondary Mathewatics 2f Southern Illinois University. The language or sets and the axioms of mathema.tics were used in the ninth grade textbook. Materials for other secondary school grades are being developed. ~ Secondary School Curriculum Committee, National Council ~ Teachers 5Z! ~athema tics. This coromi ttee made 'Natlonal Council of Teachers of Mathematics, The Revolution in School Mathematics (Washington: National Oouncil of teachers of Mathematios, 1961), p. 19. 26 studies of the mathematics curriculum and instruction in secondary schools in relation to the needs of oontemporary society. Reports of their findings could be obtained from the National Council of Teachers of Mathematics. Summary. It may be concluded that the programs discussed stressed unifying operations in mathematics. Some of the concepts were a Setslanguage and elementary theor,r Logical deductions statistical inference, probability Systems of numeration Properties of numbers Struoture Extensive use of graphical representation Valid generalizations Operations The students who attended the Tallahassee summer nathematies program at Florida State University had a mean IQ of 132, with the IQ's ranging from 122 to 152. However, the students completing the experimental program in the Tulsa Public Schools in Tulsa, Oklahoma had a mean IQ ot 119. Kennedy,6 in his program of modern mathematics for senior high school students, stated that 6Kennedy, op., cit. p. 97. 27 the text used in his class could be used for ~ senior high mathematics class. While a direct oomparison o£ the ability of the students who were completing the different programs could not be made, it may have been conoluded that any student interested in mathematics could have performed as well in the modern mathematics course as he could in a traditional course. The students who completed the work in the program under Kennedy were very successful in advanced mathematics courses at the college level. Other students who refrained from going on in mathematics were excused from taking any college mathematics because of high scores on placement tests. From the results of the three different types of programs studied, it may be concluded that a program in modern mathematics would be of help to any student who finds mathematics interesting and a personal challenge. Even the student of average ability, who likes mathematics, cangrasp and work with the concepts of modern mathematics in the areas of number systems} algebraic, logarithmic, exponential, and trigonometrio functions) operations with sets; limitS} and statistical inference and probability_ CHAPTER III PROCEDURE AND HE'fHOD OF RESEARCH The questionnaire inquiry technique which was one form of the normative survey was used in the study. The method ot research delt with an overview of the mathematics curriculum rather than with problems unique to each individual school. Procedure Questionnaire inquiries. Securing published data on experimental mathematics programs, and what had been accomplished b.Y those programs, was the first step taken toward analyzing the problem, Numerous articles were found on topics such as the need for revision in the secondary school mathematics programs, proposed changes in the mathematics ourriculum in the ~econdary school, and modern mathematics for the secondary school. It was from this material that the questionnaire was developed for a survey of the mathematics' curriculum in the secondary schools of Montana. Groups surveyed. To make the study as complete as possible, it was necessary to communicate with mathematics teachers in the 155 public high schools, .the 16 county high schools, and the 19 pl~vate high schools. This information was drawn as listed in the Directory of. Science and rlathematics  Teachers of Montana, 196263 (NDEAIII1Dll!6250 Revised). 29 On March 6, 1962, a questionnaire, a letter, (see appendix), and a stamped selfaddressed envelope were sent to one mathematics teacher, chosen at random, in each of the 190 secondary schools. Wbile there might have been some question about who was to answer the questionnaire, the writer stated that the teacher rather than the principal or other administrator was the beat qualified to give an adequate response for the following reasons. 1. It was the assumption that the teacher is the person who is in the classroom and therefore is the best judge of student performance in mathematics. 2. It was assumed that the teacher would have received the most training in mathematics and would therefore be the most qualified person to respond to the questionnaire. 3. New mathematical material is useless unless there is a competent person to instruct the students in the concepts and fundamentals. To attain a high level of proficiency in the new mathematics material that has been introduced on the market, it was assumed that the teacher 1s the one who must be constantly reading and studying in order to be able to evaluate this material and judge if it is better than the old. 30 On April 11, 1963, a duplicate questionnaire was sent out to the mathematics teachers who had not responded to the first questionnaire. A letter (see appendix) was enclosed requesting the teacher to please fill out the second questionnaire and that it was being sent because the first one had not been received. Editing 1h!. inquiries. This study was concerned with the mathematics curriculum of the secondary schools as a whole rather than with the mathematics curriculum of a partioular group of schools. Not all of the questions were answered on the questionnaires. The answers depended mostly upon the size and financial condition of the various schools which limited the extent of their rr~thematics program. Teacher knowledge of the mathematics program in the school system played ~n important part in the answering of the questionnaire. Compiled. .2!:2. Since the questionnaires had to be sent to the teaohers in the latter part of the school year eo that maximum teacherstudent relationship could be obtained, and when teacher familiarization with the mathematics program would be most complete, the percent of returns was regulated to some degree. Despite t~i8 limitation, a significant sampling of teacher responses was obtained for the study. Data acquired from 145 questionnaires was utilized in the p~eparatlon of the study. These.14S r~spon8es represented an overall return of 76.5 per cent. However, five te::lchers ret1.1rned' letters stating that they did not 31 feel qualified to answer the questionnaire as they were first year teachers in the field. Two other teachers felt that the questions were too vague and so did not fill out the questionnaire. This rr~de a total of lS2 responses or a complete total of 80 per cent response. The percentage of returns for each group of schools was as follows: (1) public senior high schools, 117/115 or 75.5 per cent, (2) county high schools, 13/16 or 81.2 per cent, (3) private high schools, 15/19 or 19 per cent. The findings of the study were recorded in the tables and in the content of the ensuine chapters. Because of the various types of problems encountered by each school in establishing a mathematical curriculum, the tables present the findings as a whole rather than by any grouping of schools. The findings of each question were aegregated for certain statistical purposes. These findings were placed in the study where they could be utilized most effectively. AnalysiS .2! ~ problem. The magnitude of the study made it necessary to sacrifice some detail for a more general dispursion or material. The four areas considered seemed to be the best in order to reflect any change toward the new mathematical philosophy that has developed and is being emphasized upon today. These areas verel 1. What 1s the general structure of the mathematics curriculum in the Montana secondary sohools? 2. What mathematical concepts are the most difficult for the studsnt to become proficient in? 3. What are the opinions ot teachers toward the mathematics curriculum? 4. What is being done to orientate elementary and secondary teachers on new mathematical concepts? 32 It was indicated that these four areas would give sufficient knowledge of the progress or lack of it in the schools, while not beooming involved in too many small details of eaoh 1ndi vidual school. SUBPROBLEMS OF THE STUDY Subproblem ! It was first necessar.y to review the general structure of the mathematics curriculum in the Montana secondar.y schools before any evaluation could be accomplished. ~ necessary. To establish the pattern of mathematical structure in the secondary schools, it was necessary to read what had been published by the state Department of Education concerning mathematical guides to the curriculum at the secondary level. Since it is common knowledge that the vast majority of all the secondary schools offer algebra, algebra II, geometry, and trigonometr, y and/or solid geometry, attention was focused upon the 33 following J (1) the number of students enrolled in each of these classesl (2) the textbooks most commonlY use~ in eaah class and when new texts would be ordered, (3) the question of whether general or business mathematics was offered and what year should it be taught, (4) the question of did ,~he school have a mathematics or mathscience club, and (5) the oriteria for an7 special mathematics program offered in the school other than the four traditional subjects? !!.ources !?!:~. The teachers provided the answers to these questions. The mathematics teacher is the closest person to the answers of these questions, and the person responsible for the mathematical training of the pupils. If the teacher is not acutely aware of the Chang~8 in the mathematics program or how the program is operated in the school, there io no justification for employing such an individual. Analysis ~~. Graphs showing the peroentage of students enrolled in each of the various mathematics olasses were plotted. A tabl.e was made showing the textbooks that were most commonly used in each class, and a graph. showing the year in which new textbooks were to Pe ordered for the various mathematics classes was included. A table of schools maintaining a mathematics club or Q. mathscience club was also included. A mathematics club is very 34 necessary to the practJ.cal application of mathematics as giving the student more opportunity to obtain help and explore mathematics on his own. There is growing concern over the problem of what to do with the noncollege bound student in regard to mathematics. The current general math classes are a dumping ground for students who can not fit in anywhere else. Many of these students are not capable ot doing anything but mechanical mathematics. Most \ of them are discouraged and afraid. This problem was discussed qui te fully while the writer was attending the Northwest Regional Conference of Mathematios Teachers at Gearhart, Oregon. It was the concern expressed about the problem of the noncollege bound etudent that prompted the writer to include the question about general mathematics, and what would be the best year for the student to pursue such a course. Space was provided for the teachers to give a resume ot any special mathemat~cs course offered in the school, whether it was for the gifted child or the slow learner. Tables were prepared of the type of program offered, and the criteria for a student to enroll in such a program. Subproblem ~ Data necessa£l. The ultimate goal of teaching is that students acquire a set of meaningful concepts that they could uee effectively to solve problems. There was sufficient evidence that in the past our students have not succeeded in acquiring these concepts, even though they master these eoncepts temporarily, to be able to pass computational examinations. In fact, good students who do highgrade work in mechanics of ltathematics often fall dOllll in quarlti ta ti vefunct,ional thinking. Source of data. To see how the students have been achieving in both the mechanics and quanti tat! ve!unctional areas of mathematics, a composite list of mathematics topics recommended by the Comrnission of Mathematios of the College Entrance Examination Board; the Illinois Curriculum Progra~ Study Group Mathematics; the Greater Cleveland Yathematic8 Program; the School Mathematics Study GrouPJ and the Ball state Teachers College Experimental Program was included in the questionnaire. Ana1lSis 2!~. The teachers ra~d the student's ability to handle the topics as most difficult, average, and easy. Whil~ this was a rather broad rating catagory, it was indicated that the teacher should have til rather definite opinion ot the difficulty experience by the students on a particular topic after seven menths of having them in the classroom. Tables were prepared to show the comparison between the mechanical and quantitativefunotional processes of mathematics as experienced . by the student. Subproblem .Q. Many volumes have been written on the philosophy, the theories on the various aspects of mathematics teaching, and the promises of teaching machines in mathematics. Along with all of this, however, the importance of the teacher's observationa and opinions cannot be overernphasized. ~ necessary. The teachers' opinions about any area of mathematics that needed improvement and how the teacher stated it could be improved. 36 Sources of data. Question twenty of the questionnaire was set aside for teacher opinions ooncerning their teaching situation, the mathematics curriculum in their sohool system, and an opinion on the revolution within their own mathematics fraternity. Analysis of~. Chapter six of this study is devoted almost entirely to teacher opinion on teaoher improvement, subject improvement, and curriculum improvement. Sub.problem D A good math~~atics program must not only include adequate teaching materials, but teachers who are prepared to teach these materials properly. \Jhat is being done to orientate both the 37 elementary and secondary teachers, so that they will have the necessar.y background to cope with mathematics programs which will soon be available  programs which must be taught in order to prepare students to be of maximum benefit to themselves and to society. ~ necessarz. It was necessa~J to find out if teachers had been attending institutes provided by the schools~ had attended summer courses in mathematics, or has been supplied with inservioe training during the school year. Sources .2f. ~. The teach.era were asked to list any course work they had taken) or if the.1 had been provided with some ~ype of institute or inservice training during the school year. Analysis 2!~. Tablet; were esiK'lblished that compared the number or teaohers who had taken part in the three different forms of teacher training: course 'Work, institute or inservice. A table was also set up to compare the number of insti tutea or inservice training programs that had been set up by the school system to orientate the elementary teachers of the system on the newer nlethoda and concepts in mathematics. SummarY. The writer had utilized the questionnaire inquiry technique which was one form of the normative survey to ,determine 38 the position ot the mathematics curriculum in the Montana secondary school with regard to the current trend of change as recommended b,y various mathematics committees including' Commission on Mathematics in their program for College Pr~paratory Mathematio€J the National Council of Teachers of MathematicsSecondary School Curriculum Committee; ~ftnesota National Laborator,y; New York State Mathematics Syllabus Committee; Bal1 State Teachers CollegeExperimental Program; and the Greater Cleveland Mathematios Program. A detailed explanation was used to define procedures which were employed ill the study. An overall of 16.5 percent of the questionnaires was recovered and recorded. This percent seems quite sufficient since a total of 80 percent of the teachers responded though not completing the questionnaire. Eight teachers stated that they did not feel qualified to answer the questionnaire as they were beginning teachers and not well acquainted with the mathematical developments. A list of the teachers who gave their consent to have their name and the name of their Bchool used in connection with this questionnaire was included in the appendix. These are the teachers who seemed the most concerned about the current problems and who were anxious to do something about it. The analysis of the problem presented the major topics of consideration. A con~lete description was given of the methods and means by wluch the basic elements of the subproblems were 39 developed. The statistical data were explicitly presented in tables and graphs. S~1tistical findings were compared when it would be beneficial to the understanding of the reader. CHAPTER IV THE GENERAL STRUCTURE OF THE MATHE}fATICS CURRICULUM The survey of the mathematics curriculum was based upon two premises) (1) that outlines of mathe~Atica1 topics recommended to be taught in the secondary sohool mathematics curriculum can be easily obtained from the State Depar~~ent of Education or from the Oommission of l1athematics and of the College Entrance Examination Board, 425 ~lest 17th Streett New York 27, New York" (2) since these outlines can be made available, a survey ot the mathematics eurr1culum with regard to: (a) the percentage of student,s attending mathematics classes, (b) textbooks used, (c) vthen schools would be ordering new textbooks, (d) the prob10ID or general mathematicswhen it sould be taught, and (e) what special mathew~tics programs are offered by the secondary schools, would enable educators in Montana in the adapting of the new mathematical recommendations to fit their particular program. THE PERCENTAGE OF STUDENTS ENROLLED IN NATHFJ'1ATICS CLASSES Breakdoltm .2f. curriculum. The ma.thematics curriculum was broken down to general I;lathematics" algebra, algebra IT" geometry, trigonometry and solid geonletry, and senior l1lat,heinatics. The 41 general mathematics course was taught in grade 9 in most instances along with algebra. Geometry liaS offered in the lOth grade and algebra II in the Ilth~ Schools offered trigonOliletr.r and solid geometry in the 12th gradee The school which have senior mathematics programs included trigonometry and solid geometry as a part of that course. Figure 1, page !~2, showed the distribution of the average percentage of the total school enrollment atten<.l:Lng matlwmatics classes. The average attendance in algebra lms 27.6 percent, the average attendance ~m6 21.9 in geometry, the average attendance was 14.7 percent in general matbematics, the average attendance was 11.45 percent in algebra II, the average att€ndanc~ ~6 7.8$ percent in t~igonometr.y and solid geometry, and 4.8 percent in senior mathematics olasses. These figures sho\1 abou't, 10 percent dropout between the lOth and 11th grades. General Mathematics. Nationally, about 65 percent of • • ....... v ninth grade students enroll in algebra, while the rest usual~ have been programed into a cottrse called general mathematios. This division has been frequently w.ade on the basis of algebra aptitude tests or other criteria applied or a~~nistered during grade eight. Montana has an average of 14.7 percent of the total school enrollment attending general mathematics classes whioh is just about the same as the national average, assuming that the typical MA THElftA TI CS CLASS General Mathematics Algebra Geometry Algebra II Trigonometry and Solid Geometry Senior Mathematics 4.8 o 10 15 20 Average percent of total enrol~nent FIGURE 1 AVERAGE PERCENTAGE OF TOTAL SCHOOL ENROLUKENT ATTENDING NA THEMATI CS CLASSES 42 27.6 30 Number of Schools 65 60 5S 50 4, 40 35 30 2$ 20 1, . 10 5 0 0 0.10 1020 20)0 Percentage of Students Enrolled FIGURE 2 NUMBER OF SCHOOLS SHOWING BREAKDOWN OR ToTAL SCHOOL ENROLLMENT ATTENDING Gf~NERAL OR COnSUMER MATHEMATICS CLASSES 43 4050 ~ 9th grade class comprises 25 percent of the total school enrollment. Figure II, page 43, illustrated the distribution of the number of schools showing the percentage of students enrolled in general mathematics. Algebra. This subject is one of the chief branches of mathematics. Mastery of mathematics depends on a sound knowledge and understanding of algebra. Montana had an average of 27.6 percent of the total school enrollment attending algebra classes which was a ver.y good percentage. One reason w~ it is so high was that many of the swall schools which have only one ~Athematics teacher have all the 9th graders enrolled in algebra instead of aome in the general mathe~AticB class. Figure III, page 45 illustrated the distribution of the number of schools showing the percentage of students enrolled in algebra. Geometry. In this subject, the student should have an informal, intuitive familiarity with simple geometric configurations. The method of geometric proof should be covered extensively along with the axioms and postulates dealing with triangles, circles, and polygone. Logic, inequalities, and both inductive and deductive reasoning should have been discussed. Geometr,r had the second highest average attendance of all Number of Schools 70 65 60 5S 50 4$ 40 3, 30 25 20 15 10 5 o o 68 010 1020 2030 3040 40,0 5060 Percentage of Students Enrolled FIGURE 3 NUNBER OF SCHOOLS SHo\VIIJG BRt""'AKDOtm OF TOTAL SCHOOL ENROLLMENT ATTENDING ALGEBFA CLASS 4, 46 the mathematics classes offered in the secondary schools with 21.9 percent. With two years of mathematics required, the difference in attendance in algebra and geometr,y was very slight. Figure IV, page 47, illustrated the distribution of the number ot schools showing the percentage of students enrolled in • geometry. Algebra II. With the two years of mathematics required behind them, there has been quite a marked deorease in the student enrollment in algebra II classes. One reason for the low average percentage was that 38 of the Montana schools (small enrollment) did not offer algebra II in their curriculum. The average percentage attendance in algebra II was 11.4" a drop of 10.45 percent from the number of students attending geometry classes. Figure V, page 48, illustrated the distribution of the number of schools showing the percentage of students enrolled in algebra II. One note: 49 schools had between 0 and 10 percent enrollment and 49 schools had between 10 and 20 percent of the total students enrolled in algebra II. Trigonometry !!!!! solid geometry. The subject matter of these courses is slowly becoming integrated with other course offerings. Trigonometry is being developed as part of the senior mathematics program while solid geometry is showing up as a small Number of Schools 60 55 50 45 40 35 30 2, 20 15 10 5 o o 2 010 1020 2030 3040 4050 5060 Percentage of Students Enrolled FIGURE 4 NUMBER OF SCHOOLS S HG~VI NG BREAKDO¥JN OF TOTAL SCHOOL ENHD1J1,!INT ATT:nn::NG GEOMETRY CLASS 47 Number of Schools 60 55 50 4$ 40 35 30 2, 20 1, 10 , o o 010 1020 2030 3040 Percentage of Students Enrolled FIGURE , NUlmER OF SCHOOW SHOWING BREAKDOWN OF TOTAL SCHOOL ENROLLMEl~T ATTENDING ALGEBRA II CLASS 48 49 part of the plane geometry course. In some cases, trigonometry is being combined with algebra II for an 11th grade course of real and complex numbers. Because of small enrollment, many of the small schools (5,) do not offer trigonometry or solid geometry, but terminate the mathematical offerings with algebra II. The average percent of students enrolled in trigonometry and solid geometry classes in the sohools that offer these courses is 7.85. This represents a 19.7S percent drop from the students that were enrolled in algebra. Figure VI,'page 50, illustrated the distribution of the number of schools showing the percentage of students enrolled in trigonometry and solid geometr,y classes. Senior mathematics. Spaoe does not permit an elaboration ot the content or pedagogic principles of teaching such principles as vectore, matrices, statistics and probability, exponential functions, set terminology, sets of ordered pairs, and Euler's formula, to name a few. Only thirtytwo schools offered a senior mathematics program and an average of 4.8 percent of the total students enrolled in senior mathematics. Table I, page 52, illustrated the criteria tor the type of program, and the criteria needed for admission to the program. Number of Schools 60 5S So 4S 40 3S 30 25 20 1, 10 5 o o o 010 1020 2030 Percentage of Students Enrolled FIGUBE 6 NUMBER OF SCHOOLS SHOWING BREAKDOWN OF TOTAL SCHOOL ENROLLHENT ATTENDING TRIGONOMETRY AND SOLID GEOrfETRY CLASSF,s $0 Number of Schools 115 110 105 100 90 85 aD 1S 10 65 60 55 50 45 40 3S 30 25 15 10 o o 010 1020 20)0 30 0 Percentage of Students Enrolled FIGtJTI.E 7 NUMBER OF SCIDOts SHO\IING BhT\l\KDOWN OF TOTAL seIDOL EUROLLHENT ATTENDI NG SENIOR LA l'HEMATICS CLASSES 51 Number of Schools 24 16 h Totals 44 TABLE I BREAX.OOWN OF ADVANCED MATHEHATICS PROGRAMS IN MONTANA SECONDARY SCHOOLS Type of eri teria for Program Admission Enrichment Acceleration Both types Grade Point Average Interest Teacher Recom .. mendation College bound National Mathematics Test Number ot Schoole 24 8 5 4 3 44 NOTE: 93 schools do not have an advanced mathematics program tor senior students. However, 32 schools are planning to adopt a program during the 1963 or 1964 school year. 53 YJATml1ATICS FOR THE NONCOLLIDE CAPABILITY (General or Business Mathematics) There is a growing concern for the students in the general mathematics category for two reasonsl First, there are many students involved in this program, and an interest should be taken in their rr~thematics education. There is much evidence that the traditional' general mathematics oourse is tailing to interest or inspire the student to achieve J in addition to the containing of inappropriate material. A booklet on this subject is available from the Educational Policy Committee of the National Council of Teachers of Mathematics. The Booklet is entitled, Disadvantaged Americans ~ Education. An examination of many of the textbooks currently in use has indicated that much of the material presented was largely 80cial in nature and written under the assumption that the general mathematics students would 1 be terminal as tar as mathematics is concerned. That this was not the case is evidenced by the fact that students are required to have two years of mathematics in high school, and that the large number of these students who enrolled in algebra at the loth grade level have little success in this subject. lwashlngton State Department of Public Instruction .. .G.....u...i..d....e...l..i..n...e.. s for Mathematics, 1962, p. 16• Number of Teachers 11 :3 12 19 11 29 Totals 14, TABLE II TEACHER OPINION ON WHEN GENERAL OR CONSUMER MATHEMATICS SHOULD BE TAUGHT IN THE SCHOOLS Grade when Student Should Take Course Drop Course 8 9 10 11 12 NOTE, See page 69 in the text. ,4 Percentage of Response 1.4 2.6 49.6 13.0 7.4 20.0 100.0 Much of the material presented in general mathematics courses was just a review of material that had been presented in 7th and 8th grade mathematics classes. The students viewed such courses with suspicion at the outset, since they realized that this was a "dressed up" version of previously experienced material that has been rather unpleasant to work with. Ninth grade students have little interest in subjects such as budgets, taxes, interest, and checking accounts as they have no encounter with them at this stage of lite. These students can become interested in measurements, simple statistical surveys built around their interests, and construction of simple devices to be used for measurement. TEXTOOOK CRITERIA In examining textbooks to see if they meet the er! teria or the Commission on Mathematics, and the School Mathematics Study Group" the following questions concerning the texts might be answered afrirmativelya2 Algebra ~ algebra II 1. Is proper and precise mathematical vooabulary used? 2. Does it present and develop the properties of the real number system, while at the same time provide 2waeh1ngton State Department of Education, OPe cit., p. 20. suffieient exercises and problems of a character which will assure the student's mastery of the manipulative skills necessary for future success? 3. Does the text contain the subject matter suggested by the Commission on Mathematics and the School ¥mthematics Study Group? 4. Does the text make it clear that the fawiliar rules of algebra are either postulates of the real number system or that they may be logically deduced trom the postulates? Are the proofs of selected theorems given, and when only intuitive agreements are given, is it made clear that these are not complete logical proofs? ,. Does the text provide sufficient motivational material so that the student is lead to see why the number system is constructed as it is and why algebraic operations are carried out at they are? 56 6.' Are ample problem lists provided so that the student can have sufficient practice to develop both the required computional and quantitivefunctional skills? 7 •. If set notation and language is introduoed, is it used conSistently tt~oughout the text material? Geometrz· 1. ' Does the beginning of the text introduce the student to the basic elements of logic? Does the text consistently refer to these concepts in the proof theorem so that they become a part of the student's mathematical thinking? 2. Are the defini tiona given meaningfully in terms of the fundamental undefined concepts of point, line and set? 3. Does the text acknowledge the necessity for all of the axioms needed for a logical development of geometry? 4. Solid geometry: Does the text contain a reasonable amount of solid geometry· included either as a seperate section or woven into the plane geometry? ,. Does the text give an introduction to coordinate geometr,r without distracting from the basic structure of the subject? These questions serve only as a guide to what material should be contained in a good textbook and to make sure that the mathematics contained therein is precise and correot. The fact that a textbook does not answer yes to these questionn does not eliminate it from being a good test. 51 Table III, page ,8, showed the distribution of the number of schools ordering textbooks at different intervals. The majority of the schools reporting ordered new textbooks very 3 to 5 years. The table showed that fiftyfive of the schools ordered new texts every 5 years and thirtyfour schools order every 4 years. Table IV, page 59, shol1ed the distribution of the number ot sohoole ordering algebra and algebra II textbooks from the years 1963 to 1967. The table showed that at least BO schools and possibly 117 schools will be ordering new algebra and algebra II textbooks after 1963. Textbooks ~ & Montall! Secondary Schools. Textbooks are an integral part of a mathematics course. Figures were drawn up showing the five or ~ix textbooks more commonly used tor each subject in the mathematics curriculum. Since general mathematics, algebra, algebra II and trigonometry, and solid geometry texts usually have the same name the course as title, just the authors were listed. In the senior mathematics course, however, the name or the book was well as the author was listed. 58 TABLE III DISTRIBUTION OF SCHOOLS WITH REGARD TO FREQUENCY OF ORDEmNG NEW MATHEMATICS TEXTBJOKS Number of Schools Ordering Textbooks 3 14 55 .31~ 16 2 1 20 NllIllber of Years Between New Texts 10 6 5 4 3 2 1 Unknown or As Needed 59 TABLE IV BREAKDOWN OF SCHOOLS ORDEP..ING NEW ALGEBRA AND ALGEBRA II TEXTBOOKS DURING THE PERIOD 1963 to 1961 ~ru:llber of Schools Textbook Year 28 Algebra 1963 24 Algebra II 196) )0 Algebra 1964 )0 Algebra II 1964 )0 Algebra 1965 25 Algebra II 196, 115, Algebra 1966 Algebra II 1966 ,7 Algebra 1967 Algebra II 1967 )6 Algebra As Needed 46 Algebra n As Needed 60 WelchonsKrickenberger is the most widely used textbook in the secondary schools. Twentynine percent of the reporting schools used this text in algebra, 46 percent used it in plane geometry, 37 percent used it in algebra II, and 49 percent used it in trigonometry and solid geometry. Twentyfive different textbooks were used in general mathematics with HartSchultS1,.;ain used by 22 percent of the reporting schools. The schools offering senior ~athematics courses used Fundamentals £f Freshman Mathematics by Allcndoerfer and Oakley, Advanced High School Hathematics by VannattaCarnahanFatmett, and ElementarY Mathematical Analysis by HuborgBristol just about the same amount, but Foundations ££ Advanced Mathe.rr.atics by KlineoBterl~ilson was used by 1~2 percent of the reporting schools offering this class, which was 27 percent above the first three books mentioned. Figures 6 through 13, pages 61 through 66, shewed the distribution of the percentage of reporting sohools using different textbooks in general mathematics, algebra" geonletry, algebra TI, trigonometry and solid geometry, and senior mathematics courses. Summa.!Z. The eurve:,r of the mathematics curriculum in the Montana secondary schools was conducted with two premises 1n minds {l) that outlines of course content can be easily obtained trom the State Department of' Education or from the Commission on Author SMSG Porter J Dunn, Allen and Gold thwai te Stein Nelson and Grime QroveMulikin.Qrove HartSchultSwain 1.2 ~l o 10 , 15 20 Percentage of Schools Using Textbook FIGURE 8 TEXTBOOKS USED BY SCHOOLS OFFERING GENERAL AND/OR CONSUMER MATH~TICS 61 25 62 Author SMSQ AkinHenderaonPingr.y SmithTottenDouglass GroveMullikin.Grove HartSchultBwain FreliehBe~~nJohnson 0.1 WelchonsKrickenberger ~27.8 o 10 20 30 Percentage of Schools Using Textbook FIGURE 9 TEXTOOOKS USED BY SCHOOLS OFFElUNG ALGEBRA Author SMSG 3 SmithUlrich Go odwinVannatta Faacett 4.6 Tully Shute8h1rkPorter HartSchul t8wain WelchonsKrickenberger o 5 10 15 20 25 30 35 40 45 Percentage of Schools Using Textbook FIGURE 10 TEXTBOOKS USED BY SCHOOLS OFFERI NG PLAIN GEOMETRY Author SIofSa 1.6 Hat,kesLubyTouton HartSchultSwain Fre11ieh~Ber.manJohn8on WelchoneKrickenberger o 5 10 15 20 25 30 35 40 Percentage of Schools Using Textbook FIGURE 11 TEXTOOOKS USED BY SCHOOLS OFFL1tlNG AWEBRA II 64 65 Author TEMAC 1.6 HooperGriswold ButlerWren Shute8hirk.Porter WelchonsKrickenberger o ~ 10 15 20 25 30 35 40 45 50 Percentage of Schools Using Textbook FIGURE 12 TEXTBOOKS USED BY SCHOOLS OFFt~RI NG TRIGONOMETRY AtID SOLID GEOMETRY Book and Author Introduction to Matrix Algebri:SMSG 3 Fundamentals of FreSF;Oiii&'thema tics AllendoerferQakley AdvanC'~~ RisE. Se hool MathematicsVannattaCarnahan Fawcett !lementn~ Mathem~tical Analysis HerborgBrIstol Foundations of Advanced l'IathematiCs KlineOSterleWilson o , 10 15 20 25 30 35 40 t~s Percentage of Schools Using Textbook FIGURE 13 TEITOOOIro USFD BY SCHOOLS OFFOUNG SENIOR MATHEfATICS 66 Mathematics and (2) that since these outlines are available, a su~ with regard to textbooks used, how often schools can order new texts, teacher opinion on general mathematics, and 67 what special programs are offered would aid in helping administrators put increased emphasis upon: 1. The underlying assui'lrptions of mathematics 2. Making the mathematical vocabular.r precise and accurate 3. Sets as a unifying concept 4. The study of both equations and inequalities 5. Comparisons of algebraic and geometric representations 6. Leading the student to discover mathematios on his own. The subject matter of elementar,y algebra is essentially unchanged. The changes being recommended by various groups as the Commission on Mathematics and the School Mathematics Study Group are the stress of the development of concepts and meanings as being equal to mechanical manipula tiona. There haa also been a change with regard to vocabulary and symbolism. The concept of set is used early in ,the course and becomes a unifying and clarifying concept t~xoughout the course.' Inequalities are developed along with equations and graphs are used much more extensively. Geometr.y is undergoing a relatively rapid evolution. Geometry is being treated in conjunction with plane geometr.y instead of as a seperate course. 66 Trigonometry is disappearing as a seperate course and appears in a senior mathematics course along with vectors, exponential functions, Euler's formula, permutations and combinations, and Cartesian coordinates. School enrollment ~ Mathematics ~ssea. The percentage of students enrolled in mathematics classes as shown by the reporting schools was ve~J good. The 27.6 pe~cent enrollment in algebra followed by the 21.9 percent in geometry tme comparable with the national average. The drop to 11.45 percent of the student enrolled in algebra II and the fact that 38 of the reporting schools do not offer algebra II was a factor to consider. Also, 113 of the 145 reporting schools did not offer a senior mathematics program and S5 of the schools did not l~ve trigonometry and solid geometry classes. Small enrollment, and only one mathematics teacher employed were some of the main reasons why more schools terminate their mathematics program with geometry or algebra II. Mathematics ~~ noncollege £~pabilitl. There was a growing concern over the students who enroll in general mathematics. This course was not a terminal one for these students as two years 69 of mathematics 1s required in the secondary sohools and much of the material content of the course was of a 80cial nature. These two factors made this course a subject of discussion and revision by mathematicians. One of the problems under discussion was the grade level at which this course should 00 offered. The teacher opinions on this topic were varied. The grade levels suggested ranged all the way from the 8th grade to the 12th grade as illustrated in Table II, page 54. Eleven teachers said that the course should be dropped from the curriculum. The majority of the teachers reporting (12 teaohers or 49 percent) said that general mathematics should be taught in the 9th grade. A number of teachers (11 or 1.4 peroent) said the 11th grade would be the appropriate level. However, 29 teachers or 20.0 percent indicated that the 12th grade would be the most appropriate. The suggested revision of the general mathematics course indicated that the 11th and 12th grades might be the best grade level to teach the social topics of this course. This was the time when the students would be most interested in such subjects as budgets, taxes, checking accounts, and interest. .T...e...x..t..b. ooks used. The type of text ueed and how often old texts were replaced was of vi tr'll importance when revising a mathematics program. Most of the schools reporting ordered new texts every 3 to 5 years. The five year program was the most popular, with $, schools ordering textbooks by this plan. The tour year plan was next with 34 schools using this plan. There was some doubt in a few schools which seemed to have no set schedule of ordering new texts, and 20 schools reported that they order as needed. There was plenty of opportunlty to acquaint teaohers and school administrators with neu texts, as 117 of the schools indicat6cl that they would be ordering 70 new algebra and algebra II textbooks aft~r the 1963 school year. Th6 textbooks authored by rlelchons and Krickenbcrger were the most widely used by the Montana schools, with 29 percent of the schools using this text in algebra, 46 percent using it in plane geometry, 37 percent in algebra II, and !~9 percent using the book in trigonometry and solid geometry. The schoole whioh reported having a senior mathematics cla.ss used the textbook Foundations 2£ ~dvanced Mathematics by KlineOsterleW11eon in 42 percent of tlw classes; a 27 percent rise above any other book. CHAPTER V MATHEMATICAL TOPICS There has been increased emphasis in the mathematics currioulum in regardsl 1. the underlying mathematical assumptions 2. precise defini tiona and vocabulary 3. sets of elements as a unifying concept 4. study of both equations and inequalities 5. comparisons of algebraic and geometric representations.1 These emphases require a new point of view for teachers, students and textbooks. Background. This chapter dealt with Bome of the mathematical topics emphasized by the current thinking. Some ideas ot presentation of these topics was included to show bow the understanding of principles can relate to computation. These examples are not to be construed as advocating a single approac h to the study of' any topic. There must be flexl. bili ty which can be attained only through many approaches. Applications are a source of interest and motivation for pupils and can aid in olarifying mathematical ideas. lBruce E. Meserve and Max A. Sobel" Mathematics for Secondar,l School Teachers (Englewood CliffsJ PrenticeHaUt Inc. 1962), p. 7. 72 Table V, page 131 showed the distribution of the number of teachers reporting the difficulties of mathematical topics for the students. Mathematical vocabulary. Much attention was being given to make sure that mathematical statements are correctly verbalized, and that words used in presenting mathematical concepts are accurate and precise. For example. some texts make the failing in that a.n informal statement which a teacher might employ orally in making an offthecuff explanation is suitable to be printed as a rule to be followedl ftTo add like fractions, or fractions with the same denominators, add the numerators." This statement taken literally, would.mean that ~ + ~ • 7 Great care should be taken to make the mathematics correct, and not to present erroneous material that must be unlearned later. Ever.y conscientious teacher should make an effort to present what can be understood b.1 his students, what will be the most useful to them, and what he believes is true. The mathematical vocabulary used by the teacher is of the utmost importance as the student must be able to use and retain this vocabulary as he progresses in mathematios. The possession of an adequate set or mental symbols corresponding to mathematical relationships must be acquired by the student. 73 TABLE V DEGREE OF DIFFICULTY OF MATHEMATICAL TOPICS FOR STUDENTS AS INDICATED BY TEACHERS OF MATHF.HATICS Degree of Difficulty Percentage By Number of Teachers Most Mathematical Topic Diffioult Most Average Easy Mathematical Vocabulary 21 65 79 14.5 Signed Numbers 16 66 63 li.O Fractions ,1 48 40 39.5 Geometric Concepts 26 83 )6 18.0 Solving Equations 9 74 62 6.2 Evaluating Formulae 31 54 60 21.4 Verbal Problems 117 18 10 81.0 Graphing 19 51 69 13.0 Solving Systems ot Equations 18 ,9 68 12.4 Fundamental Operations Addition, etc. 4 70 71 2.8 Sete" 10 21 40 13.8 Series* 23 44 69 16.0 Neatness ot Papers 9 76 60 6.2 Checking their Work 21 S2 72 14.S NOTE I if6ets and Series not taught in all schbols. This table should be read as followal Under the mathematical topio fractions, 57 teachers repor·ted that students had much dif:ficul ty with fractions, 48 teachers reported students had an average amount of trouble with fractions, and 40 teachers reported the topic was easy for the students. or the total number of teachers. 39., per cent stated that the topic was most difficult for students to master. . The reporting teachers indicated that students had a comparatively easy time assimilating mathematical vocabulary. 74 or the reporting teachers, 21 teachers reported students had difficulty with mathematioal vocabulary, 6, said that students had little difficulty understanding and using mathematical vocabular,y, and 79 teachers reported that mathematical vocabular,y was easily picked up by the students. Signed numbers. Whenever there has seemed to be a good reason to do so, mathematicians have invented new sets ot "numbers". When one auch invention is made, it i8 not hard to realize that there is no reason to stop inventing. It is easy to invent things that do no work, but hard to invent things that do work and are useful as well. As more and more sophisticated mathematics were studied, there has been occasion to use more and more sophisticated kinds of Rnumbers". In arithmetic~ subtractions like 5  8 were impossible under the system of whole numbers and fractions. To remove this limitation on subtraction, new nun!bers were created. Thus: for 5  8 .. 3 was created so the S • 8 + (3). For these new numbers, there must be names for them, symbols for them, definitions of addition and multiplication of them, and proofs that addition and multiplication of them have the usual characteristics symbolic of ad~ tion and multiplication. One of the basic defini tiona for signed numbers is, "That for every + n, a number 75 n 18 erea ted corresponding to +n by the defini ti on (+n) + (n) • 0." Thus J n is to be read nega ti ve n and is also called the addi'ti ve inverse or +n because (+n) + (n) • O. Also the number n from which +n and n are created, is the absolute value of both +n and n. While the concepts of signed numbers seem comparatively easy tor students to grasp, (only 16 ot the reporting teachers indicated that this topic was hard for students to master) students tend to neglect these signs when engaged in computational skilla. Clear understanding of the concepts ot signed numbers will help the student think through more complex mathematical situations without being cluttered mentally by errors as to the positiveness or negatlveness o~ a number. Fractions, In the system or whole numbers J there i8 no quotient for such an indicated division as 9 • 5. To remove this limitation, new numbers called fraotions were created. A fraction is an ordered pair ot whole numbers J as the fraotion 6 where b i8 not zero. The definition, i · a, makes every whole number a member ot the set of fractions. Students have little trouble multiplying tractions as: ~ • a. + sa· However, making students .see this operation in reverse i8 sometimes very difticul t. Work with fractions are built out of simple fractions and must be broken down in order to be 801 ved. If students receive practice in using the properties 76 of 1 as illustrated by the definitions _a . 1, and breaking a fractions down by reversing the multiplication process, their work in solving complex fractions would become easier. Fractions was one of the topics teachers reported students had the most difficulty with. or the teachers reporting, 57 or 39., percent reported that students experienced much difficulty in manipulation ot fractions. Forty teachers reported that fractions were easy for the students to become proficient in and 48 reported that students experienced a normal amount of difficulty in working with fraotions. Geometric concepts. One of the major concerns in the introduction of geometric concepts is careful attention to definitions. As in any logioal system, it is not possible to define everything~ Usually a point and a line are taken as undefined relations. The property of a point being on the line or plane is also taken as an undefined relation~ Then careful definitions are drawn in terms of these assumed terms and relations. In geometry, there is a necessity for precise definitions, emphasis on the nature or proof, including the significance of assumptions (postula tee and axioms). The following concepts or topics appeared in most of the newer textbooks though not necessarily as they are listed here. 1. Defini tion and history of geometry 2. Nature of a proof 3. Measurements and constructions 4. Triangles 5. Parallels 6. Inequalities, coordinates, and loci 7. Polygons 8 •. Circles 9. Areas and volumes 10. Numerical trigonometr,y ll. Deductive and induct! ve reasoning 77 Teacher opi.nion on the difficulty of each of these ooncepts was limited by the length of the questionnaire, and the fact that some of these concepts had not been reached at the time the questionnaire was sent out.· It has been the experience of the writer that student achievement in geometric concepts depends chiefly upon how well the student grasps the definitions and terminology relating to the unit. Geometric concepts were reported as difficult for the students by twentysix of the reporting teachers. Normal student difficulty was reported by eightythree teachers, and thirtysix teachers reported that geometric concepts were easy for the students to assimilate. Solving equations. When a statement is in the torm or an equation, the statement 18 interpreted to mean that the expressions on either side of the symbol, ., mean or stand for the same thing. Recently it haa been proposed that equa tiona likel ~ • 6 be called "open sentences". This was done to emphasize that, as a sentence, it is not true fot' all values of YJ that it is true for only one value and even then it may not be one of the permissable values under consideration. Solving equations was a comparatively easy concept for students to handle. Only nine of the reporting teachers indicated that this concept was very difficult for the students 78 to grasp. That the concept of solving equations was put across to the students with normal amount of effort, was reported by seventyfour teachers. Sixtytwo of the teachers reported that solving equations was an easy topic for the students to master. Evaluating Formulas. Formulas were the most useful part ot algebra to the vast majority ot people. In the presentation ot how to evaluate formulas, two alternatives present themselves. One altern~tive i8 to assume t~at the formulas that are understood most easily by students are ~hose related to geometrical figures. It appears that the C9mmission on l!athematics Ir~de the assumption that all students are familiar with geometrical figures and their formulas. 2 2Walter W. Hart and others, New First Year Algebra (Boston: D. C. Heath and Company, 1962), p. ro: The other 18 that students have a great deal or trouble mastering percentage formulae.· 79 Thougb solving equations and evaluating formulas are quite similar in mechanics of computation. thirtyone of the reporting teachers stated that students had much difficulty with formulas. This was a 15.2 percent increase over the teachers that reported much diffic\tlty in solving equations. Of the teachers responding, fiftytour reported that the students experienced a usual amount of difficulty and sixty teachers reported that evaluating formulas wae an easy concept for the students to grasp. Verbal problems. One of the most difficult parts in the teaching of rna thema tics is teac bing students to translate the works ot a verbal problem into mathematical symbols. Many devices have been reco~ended, many nspecial tricks" have been devised. The student gets the impression from these tricks and devises that there is no general approach to the solution ot verbal problems. He feels that with each type of verbal problem there 18 a particular trick or devise needed to solve the problem. The student i8 often required to transla.te verbal problems without having been well Tersed in mathematical vocabulary and t~ relationship between mathematical 6,y.mbols and words. To help students solve problems, moat text books still use a standard procedure. A type problem is given, explained by an illustrative example (usually in different colors), and followed 80 by problems involving the same teohniques for solution. Thus the student memorizes a procedure .and develops a skill, but beyond reoognition of a type of problem, does no real thinking. That students are still struggling to solve verbal problems is evidenced by the fact that 111 of the reporting teachers stated that students had most difficulty with this mathematical concept. Only 18 teachers indicated that students grasped this concept with normal effort and 10 stated that students had an easy time with verbal problems. Graphing. Graphs as an aid are at once a means of clarifying definitions, and another means of defining mathematical relations. Graphs are effective ways of presenting mathematical functions. A start is made by plotting funotions which are sets of ordered pairs of whole numbers resulting in a dot graph. Line graphs of functions may be developed later and used in the solution or systems of linear equations and in the study of inequalities. Graphical representation i8 employed to ,aid in £urniuhing the necessary instruction of systems of quadratic equations, and graphs of seta of numbers on a line. Students seem more able to express their thinking in ter.m of graphs than by translating verbal problems. Of the teaohers reporting, 19 indicated that graphing gave the students quite a bit of difficulty. Many of the teachers felt that graphing was a comparatively easy concept to grasp as fiftyseven 81 teachers reported presenting graphing to the students with a normal amount of effort, and sixtynine teachers saId that graphing was an easy concept for the students in which to achieve proficiency. Solving systems £! e~at1ons. Instruction about the solution of systems of equationsone of which is the systems of quadratic equationshaa been simplified during past years. Quadratic equations are first solved graphically to clarify the fact that there may be more than one real root and to prepare for subsequent illustrations when there might not be real roots. The procedure in drawing such a graph of quadratic equations assumes that the graph is a smooth andcontinuous curve. Though solving systems of equations is a more sophisticated procedure than solving a linear one, students still do not experience much difficulty in achieving in this ooncept. Only 18 of the teachers reported that students had an exceptional amount of difficulty with this concept. That students had to put forth a normal a~ount of effort to learn this concept was reported by 59 teachers, and 68 reported that the concept of solving systems of equations was easy for the students. Fundamental operatio~. MUch of the mect~n1cal operations of secondar,y mathematics depends upon the fQur fundamental operations of addition, ~\btraction, multiplication, and division. 82 Addi tion is in! tially defined in terms of counting and is an operation which assigns to a pair of numbers another number called the sum. Subtraction, on the other hand, may be defined in terms of addition; it is the operation for finding one addend if the sum and the other addend are known. The fact that Bubtraction is the inverse operation ot addition should not be obsoured. The conoept of roul tiplication has usuallybeen presented as repeated addition. While it is not suggested that this approach be abandoned, it could be Bupplemented by the cartesianproduct interpretation. Division should be defined in terms of multiplication just as subtraction is defined in terms of addition. Facility in computation cannot be separated from the significant contribution wAde to it by an understanding of the operations already emphasized. The pupil should aohieve a degree of maste:;:y of these operations which will enable him to think through mathematical situations without being cluttered mentally by errors. Students perform matherr~tical operations only to find that they have made an error in performing one of the fundamental operations which nlllified their result. Only four teachers reported that students had much difficulty with fundamental operations. The rest of the reporting teachers were evenly split in opinion with 70 teachers stating that students had the usual amount of difficulty, and 11 teachers stating that the fundamental operations were easy for the students. 83 ~. The concept of set 18 one of the major items being recommended for inclusion in the secondar,r school mathematios curriculum. The concept of a set is fundamental for communicating ideas in mathematics, just as it is in everyday language. Groups, herds, organizations, and teams are oollections of elements which comprise sets. Mathematics textbooks could be written without using the word set, but not without using the concept. It is claimed that this notion can be used as a unifying and clarifying concept throughout mathematics in the secondary school curriculum. The concept of a set is taken as undefined in mathematics. Sets should be thought of intuitively as collections of objects. It is possible t~ look at an object and tell whether it belongs in a set. No mention of sets should be made unless an effeotive use is made of the ~rminology and concepts in subsequent mathematical development. Not all the secondal" schools include sets in the mathematical topics considered. Of the 71 teachers reporting, 10 said that sets was a difficult topic for the students; twentyon~ said that students experienced a normal amount or difficulty, and forty said that the concepts of a set were easy for the student to understand and use. 84 Series. The concept of series was taken as the topics or sequences and progressions that appear in algebra II. A sequence of numbers is an ordered eet of numbers. It ~~plies that the numbers are written in order so that there is a first one, and each has a successor. A progression is a Bequer~e of numbers each of which, after the first one, can be obtained by combining a constant to the preceding. Series have more immediate interest for the stUdents than much of the theoretioal mathematics, because they relate to problems of ia~ediste concern to people. Also, the definition 15 ~~ed in with the subject of sets of numbers, a conformity recommended by the Co~~s8ion on Mathematics. Much of the work with series consists of mechanical manipula.tions and use of formulas. Again, according to the teachers reporting, the students do not encounter much difficulty with the mechanical operations. or the 136 teachers reporting, twentythree stated that the concept of series was difficult; fortyfour indicated that students grasped the prinoiples involved with a normal amount of effort) and sixtynine teachers reported that these concepts were easily grasped by the students. Neatness .2! papers. Students can rectify many of their errors if they can look over their papers without being confronted 8, by a mass of symbols ,and figures scattered over the paper without any apparent organization or arrangement. Teachers will also get a better idea of the pupills work if they can follow ·~he mathematioal operations from start to finish with a clear understanding of what the student is attempting to do. \ihen the paper is returned to the student, he cnn readily assertain where his mistakes occl1.red and how he can avoid them in the future, if the paper has been done in a neat and orderly manner. Montana high school students seemed to have achiG~ed some degree of success in this ~roc€ss as only 9 teachers reported difficulty with students turning in untidy papero. That students have some dlfficulty in keeping their papers presentable, was reported by seventysix teachers, and sixty teachers stated that there is no difficulty encountered with students being lax about the neatness of their rr:athematics papers. Ohecking~. Accuracy is one of the prime requisites of mathematics. Time and time again, students perfor.m all the manipulative operations of a problem" but end up with an incorrect solution because of carelessness. Much of this carelessness can be eliudnated if students will truce a little time and check their work over before handing it in. As students check over their work, they ~~ll become conscious of where they are making mistakes and tend to gain confidence in their work. 86 The students tend to have more difficulty with checking their work than with neatness. or the teachers reporting, twentyone indioated that students had much difficulty with checking their workJ fiftytwo teachers stated that the students eXi1erienced a normal amount of trouble in remembering to oheck their work; and seventytwo teaohers reported that the students were careful about checking their papers. Summary. There has been remarkable advances in the clarity and precision of mathematical discourse. PAny of the apparently diverse notions are special cases of a few underlying concepts. The conoepts presented in this chapter were an overview of topics presented throughout the secondar,y mathematics curriculum. Because sound and meaningful concepts can be just as poorly taught 8S unsound concepts J some of ·~he ourrent thinking as to the presentation of this topic in the classroom was included. A point of controversy in this chapter was what constitutes the dividing line between much difficulty, a normal amount ot effort on the part of the student, and what is easy for the student to grasp. It has been the experience of the writer that whereas the amount of difficulty of a topic for a student may be difficult to explain, a teacher can tell intuitively whether or not a topic is more difficult for the student than another topic 87 This chapter again brought out the fact that students have little difficulty doing mechanical operations as evidenced by the fact that 6.2 percent of the teachers reported much difficulty on the student's part in solving equations, and 12.4 percent in Bolving systenls of equatlvns. However, 81.0 percent of the teachers reported much student difficulty in doing verbal problems. CHAPTER VI TEACHER OPINION ON THE MATHEMATICS CURRICULUM In a study of this nature, it was neceesar,y to get teaoher opinion on what could be done to improve the mathematical structure, and their criticism as to what is wrong with the mathematical structure in Montana schools. Background. Not all teachers put down opinions and some opinions consisted of only a word or two. Because of the fact that some teachers did not want their names mentioned in oonnection with this stu~J the number of comments that could be inoluded was limited. A list of the teachers who contributed was included in the preface. The opinions were not taken verbatim but oare was taken to insure that the original meanings were not changed. The opinions were d1 vided into three groups t (1) opinions on curl·iculum, (2) comments on the elementary school program, and (3) teacher orientation. The first two groups constituted this chapter. The third group 18 contained in Chapter VII which dealt with teacher training. 89 Teacher opinions .2!l curriculum. Modern mathematics needs to be defined. Goals should be set up, and all teaohers should know what the goal are and his or her expected contribution. It does no good for a 5th grade teacher to attempt to introd~ce algebra if the 6th grade teacher is going to ignore algebra. A coordinated program is a must. Also, extension college oourses on changing mathematical needs would be a great help, as many teaohers need refresher courses in mathematics. The concepts in the grade school must be updated so that use can be made of these conoepts in the high school, instead of trying to do both at the same time. This updating should be done slowly and in coordination with all mathematics teachers from grade one to grade twelve. This would enable one hand to know what the other is doing. For this to happen, the school boards and administrators must be educated to see the advantages of change, and provide time in which the teaohers could work to make this change possible. It will be a great benefit when a tested course of study evolves out of the pres~nt revolution in mathematics. This will take time, but a course of study 1s needed especially for the senior year. The area of verbal problems should be improved becauBe this is how the students will really use their mathematics. Though fractions in the grades could be improved, the philosopqy of pushing this modern oonoept of mathematics into the lower grades is not praotical. Teaoh the students fundamentals down there. A few will be able to grasp the modern concepts, but the rest will be so confused that they will not be able to add their own grocery bill. The modern trends in mathematics should be kept in the junior high school and high school. 90 There haa been too much improper orientation of students for future work. Improvement could be made in the classification of the students for enrollment in mathematics courses. The students with poor backgrounds should be enrolled in general mathematics or remedial arithmetic; those with higher aptitudes should be placed in accelerated oourses. Most groups, espeoiallyin small sohools devote very little time to the courses of mathematics. Courses like algebra II or trigonometry are not taught at all or are taught on an alternate year basis. In some of these small schools, there is no teaoher with a mathematics major teaching mathematics. Schools have not gone to the modern mathematics program because they are not sure how to turn. Small 8chools would like to change their curriculum, but just do not knoW' how. They cannot offer tour years of mathematics straight through because ot the small student enrollment. How can some advanced concepts be taught without having algebra II first? To teach the new concepts, sets make mathematics understandable and explain wqy, instead of a lot of rules and do this or that. 91 The teacher should make sets serve as a tool of understanding. Reading and interpreting what is read plays a major part in the student's success in mathematics. Improvement should be made in the areas of reading and logic. Lack of agreement in terminology, and when to' present certain materials, among the various authors, leads to some of this hinderence. Also~ the influx or college material to the high schools before the students are rea~ to handle it, is a cause of concern. The ability to read and comprehend verbal problems is one of the major weaknesses in the ~Bthematics ability of the student. If the students could improve their ability to read, it would solve most or the problems the student has as far as mathematics is concerned. Basic fundamental operations and the translating of verbal situations into workable mathematics forms need improving. Students are slow in assimilating and correlating the total mathematics pictUl~. Vocabular,y should be stressed along with the basic operations, and the "algebraic laws" which are the basis or mathematics in school. The textbooks should emphasize the vocabular,y of mathematics and a glossary of mathematical terms and principles should be included in every textbook. Oral exercises, starting with Simple problems that demonstrate new 92 principles and progress to more difficult ones, is also needed. Most of the students have very little reaeoning power. They know most ot the "hows" but are unable to work and think in an organized manner. They, in general, 'are lacking in the very basic ideas of mathematics. "I want the answerto heck with the method." Inadequate preparation in the comprehension of written problems 1s a difficult one to correct when so many students take only two years of high school mathematics. Generally, students seem to have a good foundation in essential arithmetic processes. Modern mathematics in the high school curriculum can be appreciated only by the better students. The content and apprach of some of the newer textbooks as the School Mathematics study Group., Ball State series and others is highly approved. However, the biggest need is to change the attitudes of the teachers, students, and parents to the newer approaches to mathematios. More work needs to be done in mathematics courses for slower learners. In classes for average or below average students, the value ot traditional mathematics seems dubious, but answers to just why this 1s, are difficult to assemble. Perhaps algebraic and geometriC concepts should be introduced throughout the grade school.' Primary children seem more willing and able to grasp ideas better than high school students •. 93 Some of the valuabl~ traditional mathematics is becoming neglected tor more "sophisticated" mathematics. The mathematics curriculum oertainly needed to be revised, but for the nonmathematics major, the old algebra and trigonometry is possibly more useful. The new programs are wonderful for the exceptional student. General mathematios should be improved. Some small schools would like to revise the mathematios ourriculum but as in a case like ours, the textbooks are not worn ao~pletely away, so new texts cannot be ordered. The school board is extremely conservative~ No National Defense Education Act is allowed in the sohool. The facilities are extremely poor and no money will be spent on improvements. Finances are no worry as this is one of the more weal thy districts. General mathematics or consumer mathematics should be a more practioal course rather than a dimping ground for those students who are unable to pass algebra and geometr.y. The area of mathematics for the noncollege bound lacks textbooks and currioula context. More practical problems should be presented in the development of algebraio concepts. The modern mathematics program is a "must" for the future. Geometry is much too "old fashioned", and much of it could be either simplified or dropped from the curriculum. Textbooks at the high school level should be published 94 in the areas of logic, sets, analytical geometry, and other topics. So far, the textbooks being used are not strong in the development of the number system, the language, and use of sets. More mathematics can and should be taught at the elementary level. Most teachers have little more than six to eight hours in mathematics and few if any understand the new vocabulary. Institutes also spend too much time trying to teach over the head of the teachernot teaching the teachers anything. Hore emphasis should be placed on space relationships. Students also have difficulty working with inequalities. Verbal problems also pose a problem for most students. Most of this dirricul ty stems from the inability of the student to read and determine ~mat is being asked in the p~oblems. Geometry is in need of great improvement. The students find it difficult to comprehend the theorems and problems. There seems no way to improve instruction and make it easier for the students. There should be a nelT approach to geometry, providing intereet and stimulation to the students. Basic understanding of the various opera tiona tha t must be used in mathematics and how they all relate to each other in problem solving.. Topics such as integral domain, rings, fields, groups and set ideas are presented to the student every BO often to give the students an idea of what they will encounter in the field of mathematics.. The students bas1cal~ understand 9$ these topics but connot see any concrete use for them. They then lose interest and fail to understand many of the concepts. Theor,y in mathematics needs improvement. Modern algebra concepts are included in algebra II and the student cannot seem to get enough of it. They thoroughly enjoy advanced algebra. 9p:i.nions_ .2E. ~ elementary school 1?rogram. The primary grades is where the basis of the "modern approach" should be laid. The whole underlying foundation of our elemtntary mathematics program must be r€vised with much effort pu~ forth to replace the image of the traditional with focus on the new. E1ementa~J schools do not teach the kind of mathematics that is needed for success at the secondary level. The students don't know the "why" they do something, only the "hewn. The concepts of modern mathematics schould be introduced at the primary level and should be intensified in the junior high school. The more able students should be channeled toward the abstract concepts. The freshmen are not equipped to handle high school mathematics as a lmole. They are not capable of doing the simple operations as readily and as easily as they should be lfathema.tios needs its biggest boost from reading in the lower grades. The students have little difficulty with the major portion of the problems presented but 85 to 90 percent of them have trouble wi th verbal problems. This is shown throughout the state of Montana and the starting point for improvement in this topic is in the primary grades. More concepts should be taught in the grades besides just arithmetic. It seems that very little material in the field of geometr,y is taught in the grades. Enough algebra should be taught in the grades so that freshmen could take algebra II. This would help the students in their science courses also. The most improvement in mathematics needs to come in the elementary and junior high areas of mathematics. The junior high school students shottld either be given a course in algebra or one in insurance, taxes, banking, and other social topics. This integrated course is duplicated in the bigh school general mathematics course or in the consumer mathematics course, and is a waste of time for the better students as well as the poorer ones. Since it 18 difficult to allot time £or teaohing 96 basic arithmetical processes to high school students, a big step forward would be to improve the ari t hme tic and mathematics programs in the elementary school.. Some elementary teachers tend to skip over arithmetic or teach it in a fashion that doesn't do it justice. They tend to indoctrinate the students on uhow hard mathematics will be for them". Arithmetic and 97 mathematics at the elementary level sbould be taught in such a manner that the transition to high school and ultimately college mathematics would seem to be a natural step upward. Instead, to 9th grade algebra students, the laws of algebra seem to contradict what was taught to the students in elementary school. For example, to subtract signed numbers, the sign of the subtrahend is changed and the numbers added. Even when addition is performed, subtraction takes place if the eigne are unlike. This difficulty of negative numbers could be avoided if they were introduced in the 4th, 5th, and 6th grades. There is no reason why negative numbers cannot be understood if positive numbers are. The mathematics program in the elementary school should be revised and the mathematics program should adapt to change just as is done in any other curriculum area. However, we shouldn't depart from proven good concepts of content and methods of teaohing just for the sake of change. More rigid training should be provided in the elementary school so that students will have mastered the fundamental operations that cause most of the errors made by the students in high school mathematics. The elementary students need work with basic operations. This would enable the teacher of high school mathematics to get right to the subject matter at hand rather than spend more time on fundamental operations. This should b& accomplished in the student's grade school years. 98 The most improvement in mathematical areas should come before the student gets to high school and should come especially in the junior high school. The freshmen taking algebra just haven't had enough hard 'YTork to be prepared for algebra. Just to mention one thingfractions should be taugbb to such an extent in the grades that they would not have to be completely retaught in high school. Summary. Modern mathematics needed to be defined and goals set up so that all teachers would know what was being strived tor and what was expected of them. A seasoned course of study would help the small schools decide on how to approach the problem of devising a course ot action in mathematics. Reading and interpreting what is read plays an important part in the student's success in ma~hematics. Lack of agreement in terminology and the sequence of presentation of certain topics has contributed to a lack ot comprehension on the part of the student. It the students could improve their ability to interpret words into mathematical symbols J much of the difficul ty enoountered by the student would have disappeared. General mathematics should have been improved and not be a dumping ground for students who cannt achieve in algebra or geometry. 99 Students are interested in abstract topics in mathematics such as vectors, rings, and integral domain, but quickly lose their interest when they cannot see any conorete use for them. The elementar,y mathematics program needs revision as this is the starting plaoe for the concepts of mathematics. Elementary teachers tend to skip over arithmetic or teach it in a lackadaeical way. They tend to indootrinate the pupils on ~how hard mathematics will be for them". The fundamental operations of arithmetic should have been sufficiently presented so that they will not have to be retaught when the student reaches high school. CHAPTER VII TEACHER ORIENTATION The reorganization of mathematics in the secondar.y school posed a problem for the orientation of mathematios teachers. This orientation must include three areas of stu~: (1) modern rr~thematicsset theory, logic, and topology; (2) the need of a mathematics course for the noncollege bound student, and emphasis on the psychology of teaching this student; and (3) the procedures for the revision and the inlplementation of programs in mathematics education to meet the changing needs of society.l Background. The previous chapter mentioned a need for revision of the elementary mathematics curriculum, and a need for teacher orientation in the newer mathematical concepts. What is being done to help the elementary teacher was the main topic of this chapter. vlide reading and research seems to indicate that the elementary teacher i8 the one who is the least trained in mathematics and yet bears the brunt of the criticism for student failure in high school; still little is done to aid the elementar.y teacher in mathematics. Table Vll, page 102, showed the number of schools having lHoward F. Fehr, op. cit., p. 31. 101 Bome type of elementary teacher orientation. Teacher opinions, and a program the writer used in presenting the newer ~Bthematical ideas to the elementary achool teachers in the Columbia School District, Burbank, Washington, comprised the remainder of the chapter. Teacher opinion. If the "newU textbooks do the job expected ot them, then the area for improvement lies in the training of the teachers. Too many children are being taught that mathematics is hard. This could be prevented if the elementary teachers were trained and competent in mathematics. There are many and varied opinions about the revision in mathematics'. The textbooks are beginning to catch up, and now the problem is to get teachers that can competently teach what is presented in the textbook. Since the elementary teachers usually are the ones with the least training in mathematics, effort should be centered in that area first. There should be some organized method of making programs of teaching the new concepts of ~Athematics available to the elementary and secondary teachers in schools of small enrollment. ~ inservice course ~ elementary teachers. More and more appropriate courses in elementar,y mathematics are offered at Hontana State College and Ilontana State Uni versi ty • Teachers = Number of Schools 24 21 13 2 Total 60 TABLE VI SCHOOLS HAVING ORIENTATION PROGRA¥~ FOR ELEMENTARY TFACHP.~ ; Type of Program Inservice Institute Both Inservice and Institute other type ot Program 102 NOTE, 8S schools do not have elementary teacher orientation programs to acquaint the teachers with the new concepts and methods of teaching mathematics. 10) should attend these oourses whenever feasible. Since summer sessions are not always possible for teachers to attend, inservice training "classes" have proven to be extremely valuable and well received. The following program was used by the writer to orientate the teachers of the Columbia School District, Burbank, Washington, during the 196263 school year. The program was derived from suggestions received at the Northwest Council of Teachers of Mathematics at Gearhart, Oregon) an institute on elementary school mathematics at the University of Washington, Roy Dubisch, director) and from the inservice program presented to elementar,y teachers in the Seattle Public Schools. The program was well received by the teachers and helped the elementary teachers considerably in assimilating new mathematical concepts. MATHEi1ATICS FOR ELEHENTARY TEACHERS There has been more mathematics discovered in the past 50 years than in all time previous. People must know more in order to function effectively in the complex society of today. Are the grade school students of today going to be prepared and have sense enough to operate the society they will inherit? Our task is to provide every pupil today with the mathematical instruction which will be m.ost useful to him 104 tomorrow and make him most useful to society. 'Ever,y child should at least be given the opportunity to learn the mathematics which will enable him to achieve in a highly technological society. In the past, mathematicians have had a very lax attitude toward exposition. The readability of mathematical work was confusing, and too much use was made of such phrases as nit is obvious that", "thuB it is readily seen u, and tt1 t clearly follows". It was as though the mathematicians would be a li ttle crushed if they were perfectly understood~ There have been remarkable advances in the clarity and precision of mathematical discourse. The concept of sets is aimed at clarifying many mathematical terms that have been rather vague to the student. A sound ourriculum 1s not sufficient. Sound and meaningful concepts can be just as poorly taught as unsound concepts. A reorientation ot the mathematical program which emphasizes structural aspeots will be unseccessful unless the pedagogy is successful. Good mathematical instruction has a dynamic charaoter. Pupils should be encouraged to make conjectures and guesses, to experiment and formulate, and to understand. The teacher should make the student think and make him uncomfort
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Title  A Survey of the Mathematics Curriculum in the Montana Secondary Schools 
Description  A thesis presented to the faculty of the School of Education at Western Montana College in partial fulfillment of the requirements for the degree of Master of Science in Education 
Creator  Donovan, J. Gordon 
Genre (AAT)  Dissertations 
Type  text 
Language  eng 
Date Original  1963 
Subject (AAT)  Dissertations, Academic 
Subject (Keyword)  Dissertations 
Rights Management  http://rightsstatements.org/vocab/inCEDU/1.0 
Contributing Institution  University of Montana Western 
Geographic Coverage  Montana 
Digital Collection  Western Montana College Master of Education Theses, 19561988 
Physical Collection  Western Montana College Master of Education Theses, 19561988 
Digital Format  text/pdf 
Digitization Specifications  300dpi, PDF OCR  searchable 
Date Digitized  201310 
Transcript  A SURVEY OF THE MATHEMATICS CURRICULUM T IN THE MONTANA SECONDARY SCHOOLS A Thesis Presented to the Faculty of Western Montana College of Education In Partial Fulfillment of the Requirements for the Degree Master of Science in Education by J. Gordon Donovan September 1963 APPROVAL Advisor (. ACKNOWLEDGEMENTS The writer wishes to acknowledge the help of the following teachers; for without their assistance this study would have been impossible. Contributing teaohers Robert Graham Andrew McDermott Raymond Shackleford Glenn Thomas Darrell McCracken l1arvin Hash Gary Boyles Louis Stahl Gerald Downing Noel Teegarden Darle Hemmy James Tryon Ruth Ulmen Donald Hilla Karl Fiske Paul Aspevig Gary Evans Florence Timmerman James Muck Richard Walker Allan Hopper Sharon Robertson Edward Goodan Vernon Herbel LaVerne Frantzich Lee Von Kuster Elsie McGarvey Darrell Meskimen Sr. Marie Ferring Rev. William Allen Joseph Wolpert Cecilia Klofstad Vernon Pacovsky Donald Owen Carl Fox Sheila Wiley James McCulloh Glenn Pearson Howard McCrea Bernard MacDonald William Sweet Arlen Se~an Louis Karhi William Conners Harold Contway Arthur Baumann Kathleen Holm William Ross Joseph Cullen Dennis Olin Paul Ornberg Stanley Rasmussen Allan Skillman Thomas O'Neil Joseph Israel Ronald Steffani Paul Arneson Ronald Soiseth Phyllis Washburn Marjorie Von Bergen John Shular vIilliam Chalmers Claude Foster Harold Selvig John Oberlitner Douglas Vagg George Scott Lester Paro Edwin Goyette Donald Cole Doyle Coats Jerome Knopik Arline Hofland James Wood Delmar Klundt Walter Scott Ed! th Miller Nina Roatch Harry Baker Theodore Bergum Russell Hartford Mildred Schow Warner Fellbaum Leonard Amundson Sr. John Berchmans Leist Sr. M. Griswalda Norman Cascaden Sr. Judith Ann TABLE OF CONTENTS CHAPTER I. THE PROBLEM AND DEFINITIONS OF TIRMS USED • • • • • • PAGE 1 2 2 3 S 5 6 6 6 6 6 6 II. The Problem •••••••••• Statement of the problem ••• • • • • • • • • • • • • • • • • • • Importance of the study • Definitions of Terms Used • • • • • • • • • • • • • • • • • • • • • • • • • Secondary schools • • • • • • • • • • • • • • • • Mathematics curriculum • • • • • • • Mathematical topics • • • • • • • • • • • • • Teacher opinions •••••••••••• • • • • • • • • • • • • Senior mathematics • • • • • • • • • • • • • • • Fundamental operations Math club • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • Basic Assumptions and Limitations • • • • • • • • • 6 Assumptions • • • • • • • • • • • • • Limitations ••••••••••••• Sunur!B.ry • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • REVIEW OF THE LI TFRA TUItE • • • • • • • • • • • • • • A Comparison of the Two Philosophies of Mathematics Education • • • • • • • • • • • • • • 6 7 7 8 8 Progrants • • • • • • • • • • • • • • • • • • • • • 9 Literature on summer mathematics programs for the mathematically talented • • • • • • • • 10 CHAPTER Instructional program • • • • • • • • • • • • • • Evaluation of the program • • • • • • • • • • • • Experimental program in mathematics • • • • • • • • Prerequisites for the program • • • • • • • • • • What were some of the advantages ••• Pro blems • • • • • • • ~' • • • • • • • Modern mathematics in the senior year Problems Study Groups • • • ••• • • • • • • • • • • • • • • • • • • • • • • • • • The School Mathematics Study Group •• Uni versi ty of Illinois Curriculum Study • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • in Mathematics ••••••••• • • • • • • • University of Mar,yland Mathematics Project ••• Boston College Mathematics Institute • • • • • • • • • • Commission on Mathematics of the College Entrance Examination Board • • • • • • Developmental project in secondary mathematics Southern Illinois Universit,y The Secondary School Curriculum Committee National Council ot Teachers ot Mathematics • • • • • Sllmmary' • • • • • • • • • • • • • • • • • • • • • v PAGE 11 14 15 15 17 18 18 21 24 24 24 24 2$ 2$ 2$ 2$ 26 CHAPTER III. PROCEDURE AND METHOD OF RESEARCH • • • • • • • • • • Procedure • • • • • ••• • • • • • • • • • • • • • Questionnaire inquiries • • • • • • • • • • • • • Groups surveyed • • • • • • • • • • • • • • • • • Editing the inquiries • • • • • • • • • • • • • • Compiled data • • • • • • • • • • • • • • • ••• Analysis of the problem SubProblems of the Study • • • • • • • • • • • • • • • •• • • • • • • • • • Subproblem A • • • • • • • • • • • • • • • • 0 • Data necessary ••• • • • • • • • • • • • • • Sources of data. • • • • • • • • • • • • • • • Analysis of data Subproblem B • • Data necessary Source of data Analysis of data • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • Subproblem C • • • • • • • • • • • • • • • • • • Data necessary • • • • • • • • • • • • • • • • Sources of data • • • • • • • • • • • • • • • • Analysis of data •••••••••••••• • Subproblem D • • • • • • • • • • • • • • • • • • Data necessary • • • • • • • • • • • • • • • • Sources of data • • • • • • • • • • • ••••• Analysis of data • • • • • • • • • • • • ••• vi PAGE 28 28 28 28 )0 30 31 32 32 32 33 33 34 34 3, 35 36 36 36 36 36 37 37 37 CHAPTER IV. v. SllJll11l8.r:Y' • • • • • • • • • • • • • • • • • • • • • • THE GENERAL STRUCTURE OF THE fiLA THEMATICS . CURRICULUM • • • • • • • • • • • • • • • • • • • • The Percentage of Students Enrolled in Mathematics Classes • • • • • • • • • • • • • • • Breakdown of curriculum • • • • • • • • • • • • • • General mathematics • Algebra • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • Geometry Algebra II • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • Trigonametr,y and solid geometry • • • • • • • • • • Senior mathematics • • • • • • • • • • • • • • • • Mathematics for the noncollege capable • • • • • • Textbook Criteria • • • • • •••••••••••• Algebra and algebra II • • • • • • • • • • • • • Geometry • • • • • • • • • • • • • • • • • • • • Textbooks used by Montana aeconda~J schools • • • Summary •••••• • • • • • • • • • • • • • • • • School enrollment in mathematics classes • • • • Mathematics for the noncollege capable • • • • • Textbooks used • • • • • • • • • • • • • • • • • ~ATHEMATICAL TOPICS • • • • • • • • • • • • • •••• vii PAGE 31 40 40 40 41 44 44 46 46 49 53 " 55 56 57 60 68 68 69 11 CHAPTER viii PAG~~ Background • • • • • • • • • • • • • • • • • • • • 71 Mathematical vocabulary • • • • • • • • • • • • • • Signed numbers •••••••••••••••••• Fractions • • • • • • • • • • • • • • • • • • • • • Geometric concepts Solving equations • • • • • • • Evaluation formulas • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • 72 74~ 75 76 71 78 Verbal problems • • • • • • • • • • • • • • • • • • 79 Graphing • • • • • • • • • • • • Solving systems of equations' •• Fundamental operations ••••• • • • • • • • • • • • • • • • • • • • • • • • • • • • Sets • • • • • • • • • • • • • • • • • • • • • • • Serles • • • • • • • • • • • • • • • • • • • • • • 80 81 81 83 84 Neatness of papers ••• • • • • • • • • • • • •• 84 Checking work • • • • • • • • • • • • • • • • • • • 85 Sununary • • • • • • • • • • • • • • • • • • • • • • VI. TEACHER OPINION ON THE MATP~:MATICS CURRICULUM • • • • 86 88 88 89 95 VII. Background • • • • • • • • • • • • • • • • • • • • Teacher opinions on curriculum ••• • • • • Opinions on the elementary school program • • • • • • • • Sunnnary • • • • • • • • • • • • • • • • • • • • •• 98 TEACHER ORIENTATION • • •••• • • • • • • • • • •• 100 CHAPTF..R ix PAGE Background • • • • • • • • • • • • • • • • • • •• 100 Teacher opinion • • • • • • • • • • • • • • • • •• 101 An inservice course for elementary teachers • • • MATHEMATICS FOR ELEMENTARY TEACHERS • • • • • • • • • 101 10) Assignment I •••••••••• • • • • • • • •• 106 Problems • • • • • • • • • • • • • • • • • • •• 107 Vooabulary • • • • • • • • • • • • • • • • • • • • Nurnber . . . . . . . . . . . . . . . . . . . . ~ Numeral • • • • • • • • • • • • • • • • • • • • • Natural nurabers • • • • • • • • • • • • _ • • Q 0 Whole numbers • • • • • • • • • • • • • 108 108 108 108 108 Nonnega ti ve rational nwnbers • • • • • '. • • •• 109 As signment II • • • • • • • • •.• • • • • • • • ... 109 Problems • • • • • • • • • • • • • • • • • • • • Assignment III  • • • • • • • • •• • • • • • • • • Problems Set I' • • • • • • • • • • • • • 0 • • • Problems Set II • • • • • • • • • • • • • • • • • Problems Set n! • • • • • • • • • • • • • • • • Assignment IV • • • • • • • • • • • • • • • • • • The number one • • • • • • • • • • • • • • • • • 109 110 III 112 113 114 114 Properties of the number, one ••• • • • • • •• 114 Different names for one • • • • • • • • • • • • • The number zero • • • • • • • • • • • • • • • • • 11, llS CHAPTER Properties of the number zero • • • • • • • • • • Problems •• • • • • • • • • • • • • • • • • • • Assignment V Definitions • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • Point Space Linea Plane · . . . . ~ . . ~ . . . . . . . . . . . . • • • • • • • • • • • • • • e • • • • • • e • • • • • • • e • • • • • • • • • • • • • • • • e • • • ., ., • • eo ., Skew lines • • • • • • • • • • • • • • • • • • x PAGE 115 115 116 117 117 117 117 111 117 Ray • • • e • • • • • • • • • • • • • • • • •• 117 Angle • • • • • ~ • • ~ ~ • • • • • e • • • •• 111 Vertex Triangle Class work • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • It • • • • • • • • • • • • • • • • 111 111 117 Points to pond~r • • e • • • • • • • • • • • • ~ 119 SUlllID3.ry" • • • • • • • • • • • • • • • • • • • • •• 119 VIII. Sm~RYJ CONCLUSIONS, AND RECO~1E~~ATIONS • • e • •• 121 Summar.y and Conclusions • • • • e • • • • • • • •• 121 Need ••••••••• • • • • • • • • • • • •• 121 Problem • • Assumptions Limitations · . . . . . . . . . . . . . . . ~ . . • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • Qeneral structure of the mathematics curriculu.~ • 123 123 123 124 CHAPTER Mathematics for the noncollege Xi PAGE capability • • • • • • • • • • • • • • • • •• 12, Textbooks ••• • • • • • • • • • • • • • • • • • 126 Mathematical topios • • • • • • • • • • • • • •• 127 Teacher opinion • • • • • ~ RECOMI·1ENDATIONS ••••••• • • • • • • . . . . .. • • • • • • • • • • • Mathematics program • • • • • • • • • • • • • • • Review or 'curriculum • • • • • • • • • • • • • • Support of school author! ties • Review of starr'assignments •• • • • • • • • . •. . . . . . • • • • Program for teacher improvement • • • • • • • • • Money for mathematics education • • • . ... . BI BLIOGRAPHY •••• • • • • • • • • • • • • • APPENDIX A, LETTERS '. • • • • • • • • • • • • • APPENDIX B, QUESTIONNAIRE • • • • • • • • • • • · .• · '. • • • • 128 130 131 133 133 133 134 134 131 141 144 LIST OF TABLES TABLE PAGE I. Breakdown of Advanced Mathematics Programs in Montana Secondary Schools • • • • • • • • • • • • ~ 52 II. ,Teacher Opinion on when General or Consumer Mathematics should be Taught in the Schools • • • • •. • .. • • • • • • • • • • • • • • • ,0 III. Distribution of Schools with regard to Frequency of Ordering New Mathematics Textbooks • • • • • • • • • • • • • • • • • • • • • • IV. Breakdown of Schools Ordering New Algebra and Algebra II Textbooks during the Period . . ~ ~ . 1963 to 1967 • • • • • • • • • • • • • • • • • • • • V. Degree of Difficulty of Mathematical Topics' for Students to Handle· as Indicated by Teachers of Mathematics • • • • • • • • • • • • • Ct • VI. Schools having Orientation Programs for 54 58 59 73 Elementary Teachers • • • • • • • • • • • • • • • • 0 102 LIST OF FIGURES namm 1. Average Percentage of Total School Enrollment Attending Mathematics Classes •••• ~ • • • • • • • • • • • • • 2. Number of Schools Showing Breakdown ot Total School Enrollment ,Attending General or • • • • • Consumer Mathematics Classes • • • • • • • • • • • • • 3. Number of Schools Showing Breakdown of Total School Enrollment Attending Algebra Class • • • • • • 4. Number of Schools Showing Breakdown of Total School Enrollment Attending Geometry Class • ,. Number of Schools Showing Breakdown of Total 6. School Enrollment Attending Algebra II Class • • • • • • • • • • • • • • • • • • • Number of Schools Showing Breakdown of Total School Enrollment Attending Trigonometr,y and Solid Geometry Classes • • • • • • • • • 1. Number or Schools Showing Breakdown of To~l Sohool Enrollment Attending Senior • • • • • • • • • • • • • • • Mathematics Class • • • • • • • • • • • • • • • • • • 8. Textbooks used by Schools Offering General and/or Consumer Mathematics • • • • • • • • • • • • • PAGE 42 43 45 41 48 50 ,1 61 xiv FIGURE PAGE 9. Textbooks Used by Schools Offering Algebra • • • • • • • • • • • • • • • • • • • • • • • 62 10. Textbooks Used by Schools Offering Plain Geometry ••••• • • • • • • • • • • • • • • • 11. Textbooks Used by Schools Offering Algebra II • • • • • • • • • • • • • • • • • • • • • • 64 12. Textbooks Used by Schools Offering Trigonometr.y and Solid Geometry • • • • • • • • • • • 65 13. Textbooks Used by Schools Offering Senior Mathematics • • •• • • • • • • • • • • • • • • 66 CHAPTER I THE PROBLE'M AND DEFINITIONS OF TERMS USED Cultural changes occurring in present time have had significant implications for the curriculum as a whole and for mathematics in particular. Before the trend toward abstraction was felt, mathematics in the public school was cheitly concerned with manipulative operations and the skill acquired in performing the operations within a mathematical system rather than with an understanding of the properties of the system. The difference in point of view between the older approaoh and the modern conception was well put by Sawyer: The mathematician of older times asked, "Can I find a trick to solve this problem?" It he could not find a trick today, he looked for one tomorrow. But ••• we no longer assume that a trick need exist at all. We ask rather, "Is there any reaeon to suppose that this problem can be solved with the means we have at hand? Can it be broken up into Simpler problems? What is it that makes the problem soluble, and how can we test for aolubili ty?1t We try to discover the nature of the problem we are dealing with.l The new developments in graduate and research mathematics implied a necessary shift in emphasis for secondar.y school mathematics. The new applications of mathematics signified new problem lw. W. Sawyer, Prelude to Mathematics (Baltimore: Pelican Books, Penguin Books, Inc., 19~), p. 214. 2 material. Probability, statistical inference, finite mathematical structure, linear programwingall indicated a change in the applications and useful purposes of mathematics. The changes in the cultural, industrial, and economic patterns of many nations called for a change in educational patterns. More people should have been better trained in scientific knowledge. Even the coumon layman must have the ability to understand science in the world of today and knowing mathematics is basic to understanding science. In light of the great development in mathematics over the past thirty yea~sJ we should have examined this subject critically and carefully. The changes which those developments signify have failed to be adequately reflected ~n the present secondary school curriculum. THE PROBLEM Statement 2!..!ill! problem. It was the purpose of this study to examine the mathematics curriculum in the secondary schools of Montana and to see what improvements could be made to adjust to the new developments in mathematics. Four areas were considered, What is the general structure of the mathematics curriculum in the Montana secondary schools? What mathematical concepts are the most difficult for the student to master? r~t are the opinions of teachers toward the mathematics 3 curriculum? What is being done to orientate elementary teaohers on new mathematical concepts. Importance 2!~ stu91 The unprecedented growth of pure and applied mathematics in the United States has caused an acute shortage of good mathematicians. Supplying this demand has become a knotty problem. Mathematicians need more training than ever before. Yet, they cannot afford to spend more years in school, for mathematicians are generally most creative when very young.2 A whole new concept of mathematical education, starting as early as the third grade, has appeared as the most logical way to solve this problem. Never before have so many people applied such abstract mathematics to so great a variety of problems. To meet the demands of industry, technology, and other sciences, mathematicians have had to invent new branches of mathematics and expand old ones. They have developed new statistical methods tor controlling quality in highspeed industrial mass production. They have created an elaborate theor.y of information that enables communications engineers to evaluate precisely telephone, radio and television circuits. They have analyzed the design of automatio controls for such complicated systems as factory production lines and supersonic aircraft. 2 George A. Boehn and others, The New World of Math (New York: The Dial Press, 1959), p:51:   4 Faced with the problems of "instructing" computers what to do and how to do it, mathematicians have reopened an old and part~ dormant field} Boolean algebra. This branch of mathematics reduced the rules of logic to algebraic form. Numerical analysis, a main part of the stu~y of approximations, is another field for computers. Wi th mathematics expanding the way it has in the past few years, the need to revise the mathematics curriculum in the secondary schools has been made apparent. Young people in high sohool do not always know what careers theY' will follow. v1hat they should know beyond all doubt 1s that lack of mathematical preparation has closed man.y doorsnot only doors that open the way to engineering and the natural sciences, but also newer doors that lead to important areas of the social sciences, biological sciences, business, and industry.) Mathematics has come easier to the young. The study of some subjects may have been postponed ~dthout serious lOBS; age could have made their appreciation much 'easier. Mathematics has been differentfor the young it has been now or never. Experience has proven that high school was the place where most of our great scientists and mathematicians first acquired the interest that 3Ibid., pp. 6315. spurred them on to high aohievement. 4 Ever.y child should at least be given the opportunity to learn the mathematical concepts and skills which will make these activities accessible to him. Our task has been to provide every pupil today with the ~athematical instruction which will be most beneficial to him tomorrow and make him most useful to society. In view ot this fact and the urgency of the Situation, it becomes imperative that Montana administrators and school officials have a knowledge of the mathematics structure in the secondary schools and how mathematics teachers have reacted to the new material and concepts that have been introduced. It was hoped that this study would provide some insight into the mathematical structure of the secondar.y schools in Montana. DEFINITIONS OF TERNS USED Seconda;r schools. Public bigh schools, county high schools and private schools as designated in the Director" of Science and I~thematics Teachers of Montana, 196263; NDEAIII70~1/62,O (Revised) from the State Department of Public Instruction. 4Commission on Mathematics, Program for College preparato~ Mathematics (New York: College Entrance Examination Board, 1959, p. 12. 6 Mathematics curriculum. The mathematics courses offered in the secondar,y schools, including mathematics clubs, and extracurricular acti vi. ties dealing vIi th the subject of mathematics. Mathematical topics. This refers to the list of topics under question 1$ of the questionnaire. (See appendix). Teachers opinions. Those corr~ents which were noted by teachers in answer to question 20 of the questionnaire. (See appendix). Senior Mathematics. Advanced mathematics courses, other than trigonometry and solid geometr,y, offered in the 12th grade. Fundamental operations. The operations of addition, subtraction, multiplication and division. Math club. A student organization with the purpose to create and maintain interes~ in mathematics. BASIC ASSm1PTIONS AND LIHITATIONS Assumptions. It was assumed that the mathematical curriculum was designed for pupils of all ability levels since all students would live in the same technological culture. It was to be expected that different students would attain different levels of understanding and skill. It was assumed that the curriculum should provide materials and experiences which take 1 into account these different levels. Limitations. This study was limited to an overview of the mathematical structure in the seeondar,y schools rather than to try to deal with the multitude of different problems which would confront each individual "school. The wri tar indicated that teacher response would be best sui ted for this study. The study was limited to the one hundred ninety public, county, and private schools as designated by the director" of Science and Mathematios Teachers of Montana, 196263NDEAIII7011/6250 (Revised). Summary. The new mathematics has played a major role in the development of the complex technological world that we know today. Unfortunately for the nation's youth, this prodigious advance in mathematics education. The implementation of a revision of the mathematics curriculum has been taking place with unprecedented speed. It was the purpose of this study to examine the mathematics ourriculum in the secondary schools or Montana through means of a questionnaire on the following areas(l) general structure, (2) mathematical concepts, (3) teacher opinion, and (u) teacher orientation.to determine the degree of mathematical revision in the curriculum. CHAPTER II REVIEW OF THE LITERATURE It has been suggested that a person should study the historical development ot mathematics trom its rudimentar.y beginnings to the modern abstract form in order to gain understanding and cultural appreciation of the subject. In 1940 two separate investigating committees published reports on the ph11050PQY of mathematics. On the surface, these reports seem quite divergent. However, recent studies have confirmed that a convergence of both these philosophical surveys gives promiB.e of the best curriculum. The following chart summarizes these findings.1 A COMPARISON OF THE TWO PHILOSOPHIES OF MATHEMATICS EDUCATION: Hathematics in General Education Child Society Universe DICHOTOMY OR CONVERGENCE ORDER OF IMPORTANCE The Place of Mathematics ~ SecondarjEducation PQysical Universe Society Child lHoward F. Febr, Teaching High School Mathematics (Washington, National Education Association, 1955), p. 8. Personal living Personalsooial living Socialc1 vic rela tiona EConomiocareer relationships which call for , OBJECTIVES So01al sensitivityeethetic8 Toleraneecooperativeness Selfdirectioncreativenees Reflective thinking Formulation and solution Data Approximation Function Proof Symbolism Operation Changing values and SUBJECT MATTER PRIMARY VALUES Problemsolving ability Field p5,1chologyanalysis insight A philosophicconfigurational integrated learning program Difficul t to evalua te PRINCIPAL LEARNING ASPF:CTS TFACHING D1PLICATIONS PROGRAMS 9 Abili ty to think olearly Information, concepts, principles FUndamental skills Attitudes I nterests and appreciations Field of number Geometric form, space perception Graphio Representation Elementary analysis Logical thinking Relational thinking Symbolic representation Permanent values and Organized subject matter Association and generalization A wellstructured and organized subjectmatter program Easily evaluated Much has been written in regard to the uses or modern mathematics, the significanoe of modern mathematios for the secondary school curriculum, and the acute shortage of mathematics 10 teachers and mathematicians, both pure and applied. While much has been said about the various needs in mathematical fields, there has been very little done in the way or experimental mathematics programs tor secondary students to learn the concepts of modern mathematics. The number of programs that have been carried out are few and seem to fall into three oategories I Summer institutes, the teaching ot algebra in the eighth grade so that modern concepts could be taught during the higb school years, and programs'for twelfth grade students only. A brief summary of a progra.m in each one of these categories will be given. Literature S!!! Summer Mathematics Programs ~~ Mathematically Talented A number of summer institutes for the mathematically talented have been carried out across the nation. As all of them have been quite 8i~ilar in format and purpose, only the summer program at Florida State University, Tallahassee, Florida was outlined. The program was six weeks in duration and met for classes four to five hours daily. , The purpose of the program wass 1. To identify high school youngsters capable of becoming research mathematicians or exceptional mathematics teaohers. 2. To enhance and develop the interests of these young men and women by providing them with new insights into the expanding field of mathematical knowledge. n 3. To bring the youngsters into contact with mathematical knowledge of a kind not found in conventional high school courses, thus providing the students with opportunities to engage in creative mathematical activities. The students were divided into two groups. Group I was composed of students who had completed no more than the ninth grade by June 1959, and who had finished at least one year of algebra. Group II was composed of students who had completed the eleventh grade by June 1959 and had finished at least three years of college preparatory mathematics. Instructional Program. I. Programming the 1m! 650  3 weeks A. Introduotion 1. The memory un! t 2. The arithmetic unit 3., The control unit 4. Instruction format B. Coding 1. Arithmetical operations 2. Shirt operations 3. Branch operations 4. Watson System special operations c. Programming 1. Use of Loops 2. Setting initial condition .3. Terminal Loops 4. Speed verS\1S space D. Precision, scaling, and testing 1. Notation for fixed point scaling 2. Floating point calculations 3. Double precision calculations 4. Tracing II. Numerical Methods A. Newton's i tera ti ve method 1. Coordinate geometr.y 2. . Slope or the tangent line to a curve 3. Cube root routines B. Approximation ot areas by finite sums (Skipped for Group I) III. Problems in Programming A. Compute Phi B. Solve systems of linear equations in ten unknowns c. Compute cube roots of numbers from 1 to 10,000 to ten significant digits IV. Number theoryGroup I14 weeke. Test: Theory 2! NUmbers by Burton Jones. A. Group II 1. Integers 2. The Euclidean Algorithm 3. Greatest common divisor 4. Unique factorization into primes S. Diophantine equations 6. Congruences 1. Elementar,y properties and theorem 8. Chinese remainder theorem 9. Order 10. Fermat and Euler theorems 11. Primitive elements 12. Properties of Euler and Phi Function 13. Field axioms 114,.. Polynomials F(x) and unique factorization into irreducible polynomials 16. La Grange's theorem 12 17. Primitive roots mod 2 as applications 18. Congruence in F(x) and the proof that F(x) mod an irreducible polyno~~al is a field 19. The concepts of prime fields, extension fields, and root fields 20. The construction of projective geometries from fields 21. Elementary aspects of groups or sets 22. La Qrange~s theorem B. Group I  2 weeks 1. Integers 2. Euclidean Algorithm 3. Unique factorization into primes 4. Diophantine equations 5. Congruences 6. Chinese remainder theorem 7. Order 8. Reduced residue systems 9. Euler's theorem 10. Properties ot EUler's Phi Function 13 V. Probability and Statistics  6 weeks  Texta Introductory Probability ~ Statistical Inference~ Experimental Course. A. Organization 1. Presentation of data 2. Frequency distributions B. Summarizing a set or measurements 1. Mean of the distribution 2. Standard deviation of the distribution c. An intuitive approach to probability D. Formal approach to probability E. The laws of chance for repeated trials F. Applications of the binomial distribution and acceptance sampling and testing hypotheses G. Using samples for estimation and sampling from fim te popua tiona H. Mathel'r.a tical induction VI. Algebra  Group I and Group II  4 weeks A. Definition of a field B. Concept of a set; one to one correspondence o. Distinction between number and numeral D. Theoretic study of number systems 1. Natural numbers 2. Integers ). Rationals 4. Irrationals 5. Complex numbers E. Matrices 1. Translations, rotations, and stretohes 2. Definition and properties ot matrix mul tiplication and addition 14 3.  Solution of linear equation systems via matricies F. Symbolic logic and functions and relations 1. Statements and connective 2. Sentences and quantifers 3. Sets 4., Functions S. Relations 6. Functions and operations VII. Introduction to Russian  Group I and Group II  6 weeks  Testa Simplified Russian Grammer by Fayer, Pressman, and Pressman  )8 lessons covered. Evaluation ~ ~ program. All the objectives that were set at the beginning or the program were obtained to a high degree. The students took the STEP Mathematics Test, College Level, at the end of the summer program. The raw scores made by the students ranged from 36 to 50, and the median raw score for the group was 43 out of a maximum of 50 possible. On the College Qualification Tests, Combined Booklet Edition, the range was between 2, and 99 percent with 93 the median percentile rank for the group. 15 Group I took the Wechsler Intelligence Scale for Children and Group II took the Weohsler Adult Intelligence Scale. The range in deviation IQ' 5 was 122 to 152, with the mean IQ of the group at 132.2 Experinlent~ ProgrllIl! !!! Mathematics The Tulsa Publio Schools in Tulsa, Oklahoma started an experimental mathematios program in the spring of 1933. Two tests were given to seventh grade students near the end of that sohool year to determine placement in the experimental program. Prerequisites ~ !h! program. 1. Student must rank at the ninth grade level or above on the Arithmetic Reasoning section of the Stanford Achievement Test. 2. Student must rank about the 90th percentile on the California Algebra Aptitude Test. 3. Student must be interested in mathematics and plan to major in mathematics in high school. 2Eugene D. Nichols; "A Suw~er Mathematics Program for the Mathematically Talented, tt .!!!! Mathematics Teacher" LIll, (April, 1960), 23,. 4. Student must have written approval of the parents. 5. Student must have a reoommendation from his seventh grade teacher based on the student's IQ, work habits, and his enthusiasm for mathematical work. The average grade level of the group on the Stanford Achievement Test was 10.1 and the average on the california Algebra Aptitude Test was 93.). The average IQ was 119.0. The first three weeks of the course was devoted to an intensive study of eighth grade work. The remainder of the year was spent on traditional work in freshman algebra. The Stanford Acheivement Test, the Lankton First Year Algebra Test, and the Cooperative First Year Algebra Test were given in the spring or 1956. Average scores of the group were a8 follows: Stanlord Achievement Test 12.0 Lankton First Year Algebra Test 92.3 percent Cooperative First Year Algebra Teet 95.5 percent This was considerably higher than scores made by the regular ninth grade students. In the ninth grade, the students were given a review of basic algebra, linear equations in 1, 2 and 3 unknowns, factoring, rra~t1onal and negative exponents, quadratic equations, determinente, logarithms, imaginary and complex numbers, binomial theorem, progressions and series, powers and roots, mathematical 17 induction, variations, theory of equations, graphs of functions in 1, 2, and 3 unknowns, and probability. The average percentile rank achieved by the students on the Cooperative Advance Algebra Test, given in the spring of 1956 was 92.1. As sophomores, the group took 3/5 of a year of plane geometry and 2/, of a year of solid geometry. Their average score on the test given at the end of the session on plane geometry was 93.6 and the average score of the solid geometry test was 94.8. During their junior year, the group took trigonometry and college algebra, using a textbook in which the subjects were more or less integrated. They obtain~d an average rank of 79.6 on the Cooperative Trigonometry Test. Mathematical analysis, with some attention to probability and statistical inferenoe, was studied during the senior year. Of the original 60 pupils that started the program, 35.remained. These students consistently outscored students who had not been exposed to the advance program. Of the 24 students who dropped out, 15 had transferred to a different school. ~ ~ ~ £!. ~ advantages? 1. The students were enthusiastic about mathematics and had the necessary push to do the work required. 2. The students were capable. All the tests given indicated that they did better work than regular students 18 who were one year older. 3. They received an additional year's work in mathematios. This provided them with an adequate background tor almost any college course. 4. Because of their mathematics background, they were able to do better work in physical science courses. Problems. 1. Some of the students were not sufficiently matured and became emotionally disturbed at trivial upsets or when something went wrong. 2. There was a tendency for some of the students to feel that they belonged to a superior group. Snobbishness developed. 3. The teachers had a tendenoy to expect too much from the group in later courses. 4. Parents sometimes pressured their child beyond his capabilities. Emotional disturbances resulted.3 Modern Mathematics in the Senior Year .......      The following program was used in the twelfth grade at Wisconsin High School, Madison, Wisconsin in 19$6 and 1957. 3Coy C. Pruitt .. nAn Experimental Program in Mathematics" The Mathematics Teacher, LIll, (February, 1960), 102. 19 The text used was Principles £! Mathematics by Alendoerfer and Oakly, published by HcGrawHill Company. This text was followed closely during the entire course. While the teacher was a bit apprehensive, the students were not afraid at all and liked the idea of learning something altogether different from anything they had seen before. Logic and the number system were the first subjects tackled, and though logic and the number system are not "modern", these topics were presented from a "modern" point of view. This manner of presentation gave the students the tools and the view point necessary to study groups, fields, sets, and Boolian algebra. The students not only learned ne~ ideas but they also shook themselves free from the rigid, traditional approach to traditional algebra. They began to understand and appreciate the nature of mathematics through this new freedom. Chapter VI in the text dealt with functions and while the students had been exposed to functions in elementary algebra, it was here that they learned what function meant. This unit set the pattern and laid the foundation for the more tradi tional work that followed. The next three chapters were on algebraic funotions, trigonometric functions, and exponential and logarithmic functions. These chapters taught the students to use algebra, if nothing else, and gave the teacher a breather before proceeding into analytic 20 geometry, limits and calculus. The chapter on limits ranked along with logic and functions as the most important chapters in the course. The concept of limi t seemed to be the most difficult one for the students to grasp. Yet, they finally mastered it because they worked the hardest on that portion of material. The last chapter in the book dealt with statistics and probability. This chapter, dealing with descriptive statistics, was the most enjoyable one in the book because of its simple arithematic calculations and easily grasped conoepts. The teacher and students were fully aware at all times that their purpose was not to learn all about groups, sets, limits, statistics, and logic. The concept of taking a look at the broad field of mathematics, examining some parts in detail, and trying to learn the relationship of courses, both past and future, was fully realized. Naturally it can never be proven that a different course, traditional or otherwise, would have been of greater value. These students llere sophomores in college in 1959. Some were excused from taking any college mathematics because they scored very high on placement tests. Others, who chose mathematics for their college 'Work, began on an advanced basis. All but one of these made A'S in mathematics the first semester. They have continued to maintain high standings in college mathematics 21 courses. Some students have said that their nightly struggles with mathematics in the twelfth grade course have proved invaluable in terms of content and study habits. The teacher believes that the text used in this course could be ueed in any twelfth grade mathematics class. It would be necessar,r to omit some material and emphasize different parte of the text, depending upon the pupil's background and needs. The following problems, taken from semester examinations given during the course, show the content of the course and the side coverage of mathematical material. Problems. 1. \~ri te in "if then form": A sufficient condition that Lake Huron freeze is that the temperature be below 0 degrees F. 2. If the light is red, the ear will stop. The car does not stop. ~JNCLUSIONI The light is not red. Is this a valid conclusion? 3. !t'ind an integer x such that: x + 1 • 4 (mod 3) o~ X .c::J and 4. What real values of x satisfy 6x2 + 5x  4 :::::,01 5. Show by a series of Venn diagramst (A' fl B) I AU B' 6. Graph: 7. State the Fundamental Theorem of Algebra. 8. The inverse of the converse is the • ·9. Are identical sets equivalent? 10. In Boolean algebra (0, 1), does (1 O)X 1 11 11. rex) and g(x) are functions of x. What is g(f) called? 12. Describe the steps used in finding 53•2 b.Y logs. 13. Wri te the equa. tion for the line through (2,3) which is perpendicular to the line through (7,2) and (1,4). 14. Sn = 3  ~ is the general term of a ,sequence. Guess tne limit of the sequence and prove that this is the limit. 15. Find the first and second derivatives of rex): rex) • 8~ + j  3. 16. Find SS (3u2 + 2u)du 1 11. Calculate: mean, median, and standard deviation for this set: 5,3,3,1.9,4,6,1,2,5. 18. Prove that rex) is or is not continuous a x = ls r(x) .. ~i + x 2 , 2, X a 0 x/:o 19. vlhat is the probability of getting at least two 6t s in 2 rolls of 3 dice? 20. Prove by mathematical induction: am 4 a + ar + ar2 + • • • + arn • a  • • lr 4Joe Kennedy, "liodern Mathematics in the 12th Grade" ~ }~thematics Teacher, LII, (Februar,y, 1959), 97. 22 23 The new emphasis in the study of algebra was upon the understanding of the fundamental ideas and concepts of the subject such as the nature of number systems, ~~ctionsin particular, the linear, quadratic, exponential and logarithmic, and identities and inequalities. One way to foster an emphasis upon understanding and meaning was through the introduction of deductive reasoning. Not all reasoning is syllogistic or deductive reasoning. Training in mathematics based on deductive logic does not necessarily lead to an increased ability to argue logically in situations where insufficient data e}asted. Deductive methods are taught prima~ily to enable the pupil to learn mathematics. The subject matter contained in the courses presented was of comparatively recent development in mathematics as far as presentation of material was concerned. In the early 1940's, the study of groups and fields filtered down into the upper undergraduate years from the graduate school, and more recently, into the lower mathematios courses. Carefully seleoted material of this sort was found to be within the grasp of able bigh school students. Experience has indicated that these students have found t~~s subject matter both challenging and interesting. A course in modern mathematics would serve as an admirable means by whioh to bridge the gap between high school and college mathematics. 24 Study Groups !h! School Mathematics Study Group. This group represents the 1argest united effort for improvement in the history of mathematics education. During the school year 195960, sample textbooks and teachers' manuals for grades seven through twelve were tried out in fortyfive states by more than 400 teachers and 1,2,000 pupils. The SMSG textbooks contain new topics as well as changes in the organization and presentation of older topics. Attention was focused on important matt~mat1cal facts and skills and on basic principles that provide a logical framework for them. University of Illinois Curriculum Study ~ Mathematics. Work on the UICSM ~Aterial began in 1952 and by the 195960 school year had been used experimentally in twentyfive states by 200 teachers and 10,000 pupils. The textbooks emphasized oonsistency, precision of language, structure of matheu~tios, and understanding of basic principles through student discovery. University 2!. Maryland Mathemati.cs Projeot. The program was used in ten states by 100 teachers and 5,000 students. The courses were designed to bridge the gap between arithmetic and high school mathematics. Some of the chapter titles of the seventh grade textbook weret "Symbols", "One and Zero", 25 "Mathematical Systems", "Logic and Number Sentences."5 Boston College Mathematics Institute. Historical development is used to break away from the traditional approach and also to give the student an opportuni ty to use his Olm initiative and creativity. Mathematics was studied through problems that confronted primi ti ve If!an up through present day questions that confront mathematicians. Co~~s5ion ~MathematicB of ~ College Entrance Examination Board. The commission made recommendations looking toward the mod.ernization, modification) and improvement of the oollege preparatory mathematics curriculum in the secondary schools. The objective was to produce a curriculum suitable for students and oriented to the needs of mathematics, science, business, and industry in the second half of the 20th century. Development Project ~ Secondary Mathewatics 2f Southern Illinois University. The language or sets and the axioms of mathema.tics were used in the ninth grade textbook. Materials for other secondary school grades are being developed. ~ Secondary School Curriculum Committee, National Council ~ Teachers 5Z! ~athema tics. This coromi ttee made 'Natlonal Council of Teachers of Mathematics, The Revolution in School Mathematics (Washington: National Oouncil of teachers of Mathematios, 1961), p. 19. 26 studies of the mathematics curriculum and instruction in secondary schools in relation to the needs of oontemporary society. Reports of their findings could be obtained from the National Council of Teachers of Mathematics. Summary. It may be concluded that the programs discussed stressed unifying operations in mathematics. Some of the concepts were a Setslanguage and elementary theor,r Logical deductions statistical inference, probability Systems of numeration Properties of numbers Struoture Extensive use of graphical representation Valid generalizations Operations The students who attended the Tallahassee summer nathematies program at Florida State University had a mean IQ of 132, with the IQ's ranging from 122 to 152. However, the students completing the experimental program in the Tulsa Public Schools in Tulsa, Oklahoma had a mean IQ ot 119. Kennedy,6 in his program of modern mathematics for senior high school students, stated that 6Kennedy, op., cit. p. 97. 27 the text used in his class could be used for ~ senior high mathematics class. While a direct oomparison o£ the ability of the students who were completing the different programs could not be made, it may have been conoluded that any student interested in mathematics could have performed as well in the modern mathematics course as he could in a traditional course. The students who completed the work in the program under Kennedy were very successful in advanced mathematics courses at the college level. Other students who refrained from going on in mathematics were excused from taking any college mathematics because of high scores on placement tests. From the results of the three different types of programs studied, it may be concluded that a program in modern mathematics would be of help to any student who finds mathematics interesting and a personal challenge. Even the student of average ability, who likes mathematics, cangrasp and work with the concepts of modern mathematics in the areas of number systems} algebraic, logarithmic, exponential, and trigonometrio functions) operations with sets; limitS} and statistical inference and probability_ CHAPTER III PROCEDURE AND HE'fHOD OF RESEARCH The questionnaire inquiry technique which was one form of the normative survey was used in the study. The method ot research delt with an overview of the mathematics curriculum rather than with problems unique to each individual school. Procedure Questionnaire inquiries. Securing published data on experimental mathematics programs, and what had been accomplished b.Y those programs, was the first step taken toward analyzing the problem, Numerous articles were found on topics such as the need for revision in the secondary school mathematics programs, proposed changes in the mathematics ourriculum in the ~econdary school, and modern mathematics for the secondary school. It was from this material that the questionnaire was developed for a survey of the mathematics' curriculum in the secondary schools of Montana. Groups surveyed. To make the study as complete as possible, it was necessary to communicate with mathematics teachers in the 155 public high schools, .the 16 county high schools, and the 19 pl~vate high schools. This information was drawn as listed in the Directory of. Science and rlathematics  Teachers of Montana, 196263 (NDEAIII1Dll!6250 Revised). 29 On March 6, 1962, a questionnaire, a letter, (see appendix), and a stamped selfaddressed envelope were sent to one mathematics teacher, chosen at random, in each of the 190 secondary schools. Wbile there might have been some question about who was to answer the questionnaire, the writer stated that the teacher rather than the principal or other administrator was the beat qualified to give an adequate response for the following reasons. 1. It was the assumption that the teacher is the person who is in the classroom and therefore is the best judge of student performance in mathematics. 2. It was assumed that the teacher would have received the most training in mathematics and would therefore be the most qualified person to respond to the questionnaire. 3. New mathematical material is useless unless there is a competent person to instruct the students in the concepts and fundamentals. To attain a high level of proficiency in the new mathematics material that has been introduced on the market, it was assumed that the teacher 1s the one who must be constantly reading and studying in order to be able to evaluate this material and judge if it is better than the old. 30 On April 11, 1963, a duplicate questionnaire was sent out to the mathematics teachers who had not responded to the first questionnaire. A letter (see appendix) was enclosed requesting the teacher to please fill out the second questionnaire and that it was being sent because the first one had not been received. Editing 1h!. inquiries. This study was concerned with the mathematics curriculum of the secondary schools as a whole rather than with the mathematics curriculum of a partioular group of schools. Not all of the questions were answered on the questionnaires. The answers depended mostly upon the size and financial condition of the various schools which limited the extent of their rr~thematics program. Teacher knowledge of the mathematics program in the school system played ~n important part in the answering of the questionnaire. Compiled. .2!:2. Since the questionnaires had to be sent to the teaohers in the latter part of the school year eo that maximum teacherstudent relationship could be obtained, and when teacher familiarization with the mathematics program would be most complete, the percent of returns was regulated to some degree. Despite t~i8 limitation, a significant sampling of teacher responses was obtained for the study. Data acquired from 145 questionnaires was utilized in the p~eparatlon of the study. These.14S r~spon8es represented an overall return of 76.5 per cent. However, five te::lchers ret1.1rned' letters stating that they did not 31 feel qualified to answer the questionnaire as they were first year teachers in the field. Two other teachers felt that the questions were too vague and so did not fill out the questionnaire. This rr~de a total of lS2 responses or a complete total of 80 per cent response. The percentage of returns for each group of schools was as follows: (1) public senior high schools, 117/115 or 75.5 per cent, (2) county high schools, 13/16 or 81.2 per cent, (3) private high schools, 15/19 or 19 per cent. The findings of the study were recorded in the tables and in the content of the ensuine chapters. Because of the various types of problems encountered by each school in establishing a mathematical curriculum, the tables present the findings as a whole rather than by any grouping of schools. The findings of each question were aegregated for certain statistical purposes. These findings were placed in the study where they could be utilized most effectively. AnalysiS .2! ~ problem. The magnitude of the study made it necessary to sacrifice some detail for a more general dispursion or material. The four areas considered seemed to be the best in order to reflect any change toward the new mathematical philosophy that has developed and is being emphasized upon today. These areas verel 1. What 1s the general structure of the mathematics curriculum in the Montana secondary sohools? 2. What mathematical concepts are the most difficult for the studsnt to become proficient in? 3. What are the opinions ot teachers toward the mathematics curriculum? 4. What is being done to orientate elementary and secondary teachers on new mathematical concepts? 32 It was indicated that these four areas would give sufficient knowledge of the progress or lack of it in the schools, while not beooming involved in too many small details of eaoh 1ndi vidual school. SUBPROBLEMS OF THE STUDY Subproblem ! It was first necessar.y to review the general structure of the mathematics curriculum in the Montana secondar.y schools before any evaluation could be accomplished. ~ necessary. To establish the pattern of mathematical structure in the secondary schools, it was necessary to read what had been published by the state Department of Education concerning mathematical guides to the curriculum at the secondary level. Since it is common knowledge that the vast majority of all the secondary schools offer algebra, algebra II, geometry, and trigonometr, y and/or solid geometry, attention was focused upon the 33 following J (1) the number of students enrolled in each of these classesl (2) the textbooks most commonlY use~ in eaah class and when new texts would be ordered, (3) the question of whether general or business mathematics was offered and what year should it be taught, (4) the question of did ,~he school have a mathematics or mathscience club, and (5) the oriteria for an7 special mathematics program offered in the school other than the four traditional subjects? !!.ources !?!:~. The teachers provided the answers to these questions. The mathematics teacher is the closest person to the answers of these questions, and the person responsible for the mathematical training of the pupils. If the teacher is not acutely aware of the Chang~8 in the mathematics program or how the program is operated in the school, there io no justification for employing such an individual. Analysis ~~. Graphs showing the peroentage of students enrolled in each of the various mathematics olasses were plotted. A tabl.e was made showing the textbooks that were most commonly used in each class, and a graph. showing the year in which new textbooks were to Pe ordered for the various mathematics classes was included. A table of schools maintaining a mathematics club or Q. mathscience club was also included. A mathematics club is very 34 necessary to the practJ.cal application of mathematics as giving the student more opportunity to obtain help and explore mathematics on his own. There is growing concern over the problem of what to do with the noncollege bound student in regard to mathematics. The current general math classes are a dumping ground for students who can not fit in anywhere else. Many of these students are not capable ot doing anything but mechanical mathematics. Most \ of them are discouraged and afraid. This problem was discussed qui te fully while the writer was attending the Northwest Regional Conference of Mathematios Teachers at Gearhart, Oregon. It was the concern expressed about the problem of the noncollege bound etudent that prompted the writer to include the question about general mathematics, and what would be the best year for the student to pursue such a course. Space was provided for the teachers to give a resume ot any special mathemat~cs course offered in the school, whether it was for the gifted child or the slow learner. Tables were prepared of the type of program offered, and the criteria for a student to enroll in such a program. Subproblem ~ Data necessa£l. The ultimate goal of teaching is that students acquire a set of meaningful concepts that they could uee effectively to solve problems. There was sufficient evidence that in the past our students have not succeeded in acquiring these concepts, even though they master these eoncepts temporarily, to be able to pass computational examinations. In fact, good students who do highgrade work in mechanics of ltathematics often fall dOllll in quarlti ta ti vefunct,ional thinking. Source of data. To see how the students have been achieving in both the mechanics and quanti tat! ve!unctional areas of mathematics, a composite list of mathematics topics recommended by the Comrnission of Mathematios of the College Entrance Examination Board; the Illinois Curriculum Progra~ Study Group Mathematics; the Greater Cleveland Yathematic8 Program; the School Mathematics Study GrouPJ and the Ball state Teachers College Experimental Program was included in the questionnaire. Ana1lSis 2!~. The teachers ra~d the student's ability to handle the topics as most difficult, average, and easy. Whil~ this was a rather broad rating catagory, it was indicated that the teacher should have til rather definite opinion ot the difficulty experience by the students on a particular topic after seven menths of having them in the classroom. Tables were prepared to show the comparison between the mechanical and quantitativefunotional processes of mathematics as experienced . by the student. Subproblem .Q. Many volumes have been written on the philosophy, the theories on the various aspects of mathematics teaching, and the promises of teaching machines in mathematics. Along with all of this, however, the importance of the teacher's observationa and opinions cannot be overernphasized. ~ necessary. The teachers' opinions about any area of mathematics that needed improvement and how the teacher stated it could be improved. 36 Sources of data. Question twenty of the questionnaire was set aside for teacher opinions ooncerning their teaching situation, the mathematics curriculum in their sohool system, and an opinion on the revolution within their own mathematics fraternity. Analysis of~. Chapter six of this study is devoted almost entirely to teacher opinion on teaoher improvement, subject improvement, and curriculum improvement. Sub.problem D A good math~~atics program must not only include adequate teaching materials, but teachers who are prepared to teach these materials properly. \Jhat is being done to orientate both the 37 elementary and secondary teachers, so that they will have the necessar.y background to cope with mathematics programs which will soon be available  programs which must be taught in order to prepare students to be of maximum benefit to themselves and to society. ~ necessarz. It was necessa~J to find out if teachers had been attending institutes provided by the schools~ had attended summer courses in mathematics, or has been supplied with inservioe training during the school year. Sources .2f. ~. The teach.era were asked to list any course work they had taken) or if the.1 had been provided with some ~ype of institute or inservice training during the school year. Analysis 2!~. Tablet; were esiK'lblished that compared the number or teaohers who had taken part in the three different forms of teacher training: course 'Work, institute or inservice. A table was also set up to compare the number of insti tutea or inservice training programs that had been set up by the school system to orientate the elementary teachers of the system on the newer nlethoda and concepts in mathematics. SummarY. The writer had utilized the questionnaire inquiry technique which was one form of the normative survey to ,determine 38 the position ot the mathematics curriculum in the Montana secondary school with regard to the current trend of change as recommended b,y various mathematics committees including' Commission on Mathematics in their program for College Pr~paratory Mathematio€J the National Council of Teachers of MathematicsSecondary School Curriculum Committee; ~ftnesota National Laborator,y; New York State Mathematics Syllabus Committee; Bal1 State Teachers CollegeExperimental Program; and the Greater Cleveland Mathematios Program. A detailed explanation was used to define procedures which were employed ill the study. An overall of 16.5 percent of the questionnaires was recovered and recorded. This percent seems quite sufficient since a total of 80 percent of the teachers responded though not completing the questionnaire. Eight teachers stated that they did not feel qualified to answer the questionnaire as they were beginning teachers and not well acquainted with the mathematical developments. A list of the teachers who gave their consent to have their name and the name of their Bchool used in connection with this questionnaire was included in the appendix. These are the teachers who seemed the most concerned about the current problems and who were anxious to do something about it. The analysis of the problem presented the major topics of consideration. A con~lete description was given of the methods and means by wluch the basic elements of the subproblems were 39 developed. The statistical data were explicitly presented in tables and graphs. S~1tistical findings were compared when it would be beneficial to the understanding of the reader. CHAPTER IV THE GENERAL STRUCTURE OF THE MATHE}fATICS CURRICULUM The survey of the mathematics curriculum was based upon two premises) (1) that outlines of mathe~Atica1 topics recommended to be taught in the secondary sohool mathematics curriculum can be easily obtained from the State Depar~~ent of Education or from the Oommission of l1athematics and of the College Entrance Examination Board, 425 ~lest 17th Streett New York 27, New York" (2) since these outlines can be made available, a survey ot the mathematics eurr1culum with regard to: (a) the percentage of student,s attending mathematics classes, (b) textbooks used, (c) vthen schools would be ordering new textbooks, (d) the prob10ID or general mathematicswhen it sould be taught, and (e) what special mathew~tics programs are offered by the secondary schools, would enable educators in Montana in the adapting of the new mathematical recommendations to fit their particular program. THE PERCENTAGE OF STUDENTS ENROLLED IN NATHFJ'1ATICS CLASSES Breakdoltm .2f. curriculum. The ma.thematics curriculum was broken down to general I;lathematics" algebra, algebra IT" geometry, trigonometry and solid geonletry, and senior l1lat,heinatics. The 41 general mathematics course was taught in grade 9 in most instances along with algebra. Geometry liaS offered in the lOth grade and algebra II in the Ilth~ Schools offered trigonOliletr.r and solid geometry in the 12th gradee The school which have senior mathematics programs included trigonometry and solid geometry as a part of that course. Figure 1, page !~2, showed the distribution of the average percentage of the total school enrollment atten<.l:Lng matlwmatics classes. The average attendance in algebra lms 27.6 percent, the average attendance ~m6 21.9 in geometry, the average attendance was 14.7 percent in general matbematics, the average attendance was 11.45 percent in algebra II, the average att€ndanc~ ~6 7.8$ percent in t~igonometr.y and solid geometry, and 4.8 percent in senior mathematics olasses. These figures sho\1 abou't, 10 percent dropout between the lOth and 11th grades. General Mathematics. Nationally, about 65 percent of • • ....... v ninth grade students enroll in algebra, while the rest usual~ have been programed into a cottrse called general mathematios. This division has been frequently w.ade on the basis of algebra aptitude tests or other criteria applied or a~~nistered during grade eight. Montana has an average of 14.7 percent of the total school enrollment attending general mathematics classes whioh is just about the same as the national average, assuming that the typical MA THElftA TI CS CLASS General Mathematics Algebra Geometry Algebra II Trigonometry and Solid Geometry Senior Mathematics 4.8 o 10 15 20 Average percent of total enrol~nent FIGURE 1 AVERAGE PERCENTAGE OF TOTAL SCHOOL ENROLUKENT ATTENDING NA THEMATI CS CLASSES 42 27.6 30 Number of Schools 65 60 5S 50 4, 40 35 30 2$ 20 1, . 10 5 0 0 0.10 1020 20)0 Percentage of Students Enrolled FIGURE 2 NUMBER OF SCHOOLS SHOWING BREAKDOWN OR ToTAL SCHOOL ENROLLMENT ATTENDING Gf~NERAL OR COnSUMER MATHEMATICS CLASSES 43 4050 ~ 9th grade class comprises 25 percent of the total school enrollment. Figure II, page 43, illustrated the distribution of the number of schools showing the percentage of students enrolled in general mathematics. Algebra. This subject is one of the chief branches of mathematics. Mastery of mathematics depends on a sound knowledge and understanding of algebra. Montana had an average of 27.6 percent of the total school enrollment attending algebra classes which was a ver.y good percentage. One reason w~ it is so high was that many of the swall schools which have only one ~Athematics teacher have all the 9th graders enrolled in algebra instead of aome in the general mathe~AticB class. Figure III, page 45 illustrated the distribution of the number of schools showing the percentage of students enrolled in algebra. Geometry. In this subject, the student should have an informal, intuitive familiarity with simple geometric configurations. The method of geometric proof should be covered extensively along with the axioms and postulates dealing with triangles, circles, and polygone. Logic, inequalities, and both inductive and deductive reasoning should have been discussed. Geometr,r had the second highest average attendance of all Number of Schools 70 65 60 5S 50 4$ 40 3, 30 25 20 15 10 5 o o 68 010 1020 2030 3040 40,0 5060 Percentage of Students Enrolled FIGURE 3 NUNBER OF SCHOOLS SHo\VIIJG BRt""'AKDOtm OF TOTAL SCHOOL ENROLLMENT ATTENDING ALGEBFA CLASS 4, 46 the mathematics classes offered in the secondary schools with 21.9 percent. With two years of mathematics required, the difference in attendance in algebra and geometr,y was very slight. Figure IV, page 47, illustrated the distribution of the number ot schools showing the percentage of students enrolled in • geometry. Algebra II. With the two years of mathematics required behind them, there has been quite a marked deorease in the student enrollment in algebra II classes. One reason for the low average percentage was that 38 of the Montana schools (small enrollment) did not offer algebra II in their curriculum. The average percentage attendance in algebra II was 11.4" a drop of 10.45 percent from the number of students attending geometry classes. Figure V, page 48, illustrated the distribution of the number of schools showing the percentage of students enrolled in algebra II. One note: 49 schools had between 0 and 10 percent enrollment and 49 schools had between 10 and 20 percent of the total students enrolled in algebra II. Trigonometry !!!!! solid geometry. The subject matter of these courses is slowly becoming integrated with other course offerings. Trigonometry is being developed as part of the senior mathematics program while solid geometry is showing up as a small Number of Schools 60 55 50 45 40 35 30 2, 20 15 10 5 o o 2 010 1020 2030 3040 4050 5060 Percentage of Students Enrolled FIGURE 4 NUMBER OF SCHOOLS S HG~VI NG BREAKDO¥JN OF TOTAL SCHOOL ENHD1J1,!INT ATT:nn::NG GEOMETRY CLASS 47 Number of Schools 60 55 50 4$ 40 35 30 2, 20 1, 10 , o o 010 1020 2030 3040 Percentage of Students Enrolled FIGURE , NUlmER OF SCHOOW SHOWING BREAKDOWN OF TOTAL SCHOOL ENROLLMEl~T ATTENDING ALGEBRA II CLASS 48 49 part of the plane geometry course. In some cases, trigonometry is being combined with algebra II for an 11th grade course of real and complex numbers. Because of small enrollment, many of the small schools (5,) do not offer trigonometry or solid geometry, but terminate the mathematical offerings with algebra II. The average percent of students enrolled in trigonometry and solid geometry classes in the sohools that offer these courses is 7.85. This represents a 19.7S percent drop from the students that were enrolled in algebra. Figure VI,'page 50, illustrated the distribution of the number of schools showing the percentage of students enrolled in trigonometry and solid geometr,y classes. Senior mathematics. Spaoe does not permit an elaboration ot the content or pedagogic principles of teaching such principles as vectore, matrices, statistics and probability, exponential functions, set terminology, sets of ordered pairs, and Euler's formula, to name a few. Only thirtytwo schools offered a senior mathematics program and an average of 4.8 percent of the total students enrolled in senior mathematics. Table I, page 52, illustrated the criteria tor the type of program, and the criteria needed for admission to the program. Number of Schools 60 5S So 4S 40 3S 30 25 20 1, 10 5 o o o 010 1020 2030 Percentage of Students Enrolled FIGUBE 6 NUMBER OF SCHOOLS SHOWING BREAKDOWN OF TOTAL SCHOOL ENROLLHENT ATTENDING TRIGONOMETRY AND SOLID GEOrfETRY CLASSF,s $0 Number of Schools 115 110 105 100 90 85 aD 1S 10 65 60 55 50 45 40 3S 30 25 15 10 o o 010 1020 20)0 30 0 Percentage of Students Enrolled FIGtJTI.E 7 NUMBER OF SCIDOts SHO\IING BhT\l\KDOWN OF TOTAL seIDOL EUROLLHENT ATTENDI NG SENIOR LA l'HEMATICS CLASSES 51 Number of Schools 24 16 h Totals 44 TABLE I BREAX.OOWN OF ADVANCED MATHEHATICS PROGRAMS IN MONTANA SECONDARY SCHOOLS Type of eri teria for Program Admission Enrichment Acceleration Both types Grade Point Average Interest Teacher Recom .. mendation College bound National Mathematics Test Number ot Schoole 24 8 5 4 3 44 NOTE: 93 schools do not have an advanced mathematics program tor senior students. However, 32 schools are planning to adopt a program during the 1963 or 1964 school year. 53 YJATml1ATICS FOR THE NONCOLLIDE CAPABILITY (General or Business Mathematics) There is a growing concern for the students in the general mathematics category for two reasonsl First, there are many students involved in this program, and an interest should be taken in their rr~thematics education. There is much evidence that the traditional' general mathematics oourse is tailing to interest or inspire the student to achieve J in addition to the containing of inappropriate material. A booklet on this subject is available from the Educational Policy Committee of the National Council of Teachers of Mathematics. The Booklet is entitled, Disadvantaged Americans ~ Education. An examination of many of the textbooks currently in use has indicated that much of the material presented was largely 80cial in nature and written under the assumption that the general mathematics students would 1 be terminal as tar as mathematics is concerned. That this was not the case is evidenced by the fact that students are required to have two years of mathematics in high school, and that the large number of these students who enrolled in algebra at the loth grade level have little success in this subject. lwashlngton State Department of Public Instruction .. .G.....u...i..d....e...l..i..n...e.. s for Mathematics, 1962, p. 16• Number of Teachers 11 :3 12 19 11 29 Totals 14, TABLE II TEACHER OPINION ON WHEN GENERAL OR CONSUMER MATHEMATICS SHOULD BE TAUGHT IN THE SCHOOLS Grade when Student Should Take Course Drop Course 8 9 10 11 12 NOTE, See page 69 in the text. ,4 Percentage of Response 1.4 2.6 49.6 13.0 7.4 20.0 100.0 Much of the material presented in general mathematics courses was just a review of material that had been presented in 7th and 8th grade mathematics classes. The students viewed such courses with suspicion at the outset, since they realized that this was a "dressed up" version of previously experienced material that has been rather unpleasant to work with. Ninth grade students have little interest in subjects such as budgets, taxes, interest, and checking accounts as they have no encounter with them at this stage of lite. These students can become interested in measurements, simple statistical surveys built around their interests, and construction of simple devices to be used for measurement. TEXTOOOK CRITERIA In examining textbooks to see if they meet the er! teria or the Commission on Mathematics, and the School Mathematics Study Group" the following questions concerning the texts might be answered afrirmativelya2 Algebra ~ algebra II 1. Is proper and precise mathematical vooabulary used? 2. Does it present and develop the properties of the real number system, while at the same time provide 2waeh1ngton State Department of Education, OPe cit., p. 20. suffieient exercises and problems of a character which will assure the student's mastery of the manipulative skills necessary for future success? 3. Does the text contain the subject matter suggested by the Commission on Mathematics and the School ¥mthematics Study Group? 4. Does the text make it clear that the fawiliar rules of algebra are either postulates of the real number system or that they may be logically deduced trom the postulates? Are the proofs of selected theorems given, and when only intuitive agreements are given, is it made clear that these are not complete logical proofs? ,. Does the text provide sufficient motivational material so that the student is lead to see why the number system is constructed as it is and why algebraic operations are carried out at they are? 56 6.' Are ample problem lists provided so that the student can have sufficient practice to develop both the required computional and quantitivefunctional skills? 7 •. If set notation and language is introduoed, is it used conSistently tt~oughout the text material? Geometrz· 1. ' Does the beginning of the text introduce the student to the basic elements of logic? Does the text consistently refer to these concepts in the proof theorem so that they become a part of the student's mathematical thinking? 2. Are the defini tiona given meaningfully in terms of the fundamental undefined concepts of point, line and set? 3. Does the text acknowledge the necessity for all of the axioms needed for a logical development of geometry? 4. Solid geometry: Does the text contain a reasonable amount of solid geometry· included either as a seperate section or woven into the plane geometry? ,. Does the text give an introduction to coordinate geometr,r without distracting from the basic structure of the subject? These questions serve only as a guide to what material should be contained in a good textbook and to make sure that the mathematics contained therein is precise and correot. The fact that a textbook does not answer yes to these questionn does not eliminate it from being a good test. 51 Table III, page ,8, showed the distribution of the number of schools ordering textbooks at different intervals. The majority of the schools reporting ordered new textbooks very 3 to 5 years. The table showed that fiftyfive of the schools ordered new texts every 5 years and thirtyfour schools order every 4 years. Table IV, page 59, shol1ed the distribution of the number ot sohoole ordering algebra and algebra II textbooks from the years 1963 to 1967. The table showed that at least BO schools and possibly 117 schools will be ordering new algebra and algebra II textbooks after 1963. Textbooks ~ & Montall! Secondary Schools. Textbooks are an integral part of a mathematics course. Figures were drawn up showing the five or ~ix textbooks more commonly used tor each subject in the mathematics curriculum. Since general mathematics, algebra, algebra II and trigonometry, and solid geometry texts usually have the same name the course as title, just the authors were listed. In the senior mathematics course, however, the name or the book was well as the author was listed. 58 TABLE III DISTRIBUTION OF SCHOOLS WITH REGARD TO FREQUENCY OF ORDEmNG NEW MATHEMATICS TEXTBJOKS Number of Schools Ordering Textbooks 3 14 55 .31~ 16 2 1 20 NllIllber of Years Between New Texts 10 6 5 4 3 2 1 Unknown or As Needed 59 TABLE IV BREAKDOWN OF SCHOOLS ORDEP..ING NEW ALGEBRA AND ALGEBRA II TEXTBOOKS DURING THE PERIOD 1963 to 1961 ~ru:llber of Schools Textbook Year 28 Algebra 1963 24 Algebra II 196) )0 Algebra 1964 )0 Algebra II 1964 )0 Algebra 1965 25 Algebra II 196, 115, Algebra 1966 Algebra II 1966 ,7 Algebra 1967 Algebra II 1967 )6 Algebra As Needed 46 Algebra n As Needed 60 WelchonsKrickenberger is the most widely used textbook in the secondary schools. Twentynine percent of the reporting schools used this text in algebra, 46 percent used it in plane geometry, 37 percent used it in algebra II, and 49 percent used it in trigonometry and solid geometry. Twentyfive different textbooks were used in general mathematics with HartSchultS1,.;ain used by 22 percent of the reporting schools. The schools offering senior ~athematics courses used Fundamentals £f Freshman Mathematics by Allcndoerfer and Oakley, Advanced High School Hathematics by VannattaCarnahanFatmett, and ElementarY Mathematical Analysis by HuborgBristol just about the same amount, but Foundations ££ Advanced Mathe.rr.atics by KlineoBterl~ilson was used by 1~2 percent of the reporting schools offering this class, which was 27 percent above the first three books mentioned. Figures 6 through 13, pages 61 through 66, shewed the distribution of the percentage of reporting sohools using different textbooks in general mathematics, algebra" geonletry, algebra TI, trigonometry and solid geometry, and senior mathematics courses. Summa.!Z. The eurve:,r of the mathematics curriculum in the Montana secondary schools was conducted with two premises 1n minds {l) that outlines of course content can be easily obtained trom the State Department of' Education or from the Commission on Author SMSG Porter J Dunn, Allen and Gold thwai te Stein Nelson and Grime QroveMulikin.Qrove HartSchultSwain 1.2 ~l o 10 , 15 20 Percentage of Schools Using Textbook FIGURE 8 TEXTBOOKS USED BY SCHOOLS OFFERING GENERAL AND/OR CONSUMER MATH~TICS 61 25 62 Author SMSQ AkinHenderaonPingr.y SmithTottenDouglass GroveMullikin.Grove HartSchultBwain FreliehBe~~nJohnson 0.1 WelchonsKrickenberger ~27.8 o 10 20 30 Percentage of Schools Using Textbook FIGURE 9 TEXTOOOKS USED BY SCHOOLS OFFElUNG ALGEBRA Author SMSG 3 SmithUlrich Go odwinVannatta Faacett 4.6 Tully Shute8h1rkPorter HartSchul t8wain WelchonsKrickenberger o 5 10 15 20 25 30 35 40 45 Percentage of Schools Using Textbook FIGURE 10 TEXTBOOKS USED BY SCHOOLS OFFERI NG PLAIN GEOMETRY Author SIofSa 1.6 Hat,kesLubyTouton HartSchultSwain Fre11ieh~Ber.manJohn8on WelchoneKrickenberger o 5 10 15 20 25 30 35 40 Percentage of Schools Using Textbook FIGURE 11 TEXTOOOKS USED BY SCHOOLS OFFL1tlNG AWEBRA II 64 65 Author TEMAC 1.6 HooperGriswold ButlerWren Shute8hirk.Porter WelchonsKrickenberger o ~ 10 15 20 25 30 35 40 45 50 Percentage of Schools Using Textbook FIGURE 12 TEXTBOOKS USED BY SCHOOLS OFFt~RI NG TRIGONOMETRY AtID SOLID GEOMETRY Book and Author Introduction to Matrix Algebri:SMSG 3 Fundamentals of FreSF;Oiii&'thema tics AllendoerferQakley AdvanC'~~ RisE. Se hool MathematicsVannattaCarnahan Fawcett !lementn~ Mathem~tical Analysis HerborgBrIstol Foundations of Advanced l'IathematiCs KlineOSterleWilson o , 10 15 20 25 30 35 40 t~s Percentage of Schools Using Textbook FIGURE 13 TEITOOOIro USFD BY SCHOOLS OFFOUNG SENIOR MATHEfATICS 66 Mathematics and (2) that since these outlines are available, a su~ with regard to textbooks used, how often schools can order new texts, teacher opinion on general mathematics, and 67 what special programs are offered would aid in helping administrators put increased emphasis upon: 1. The underlying assui'lrptions of mathematics 2. Making the mathematical vocabular.r precise and accurate 3. Sets as a unifying concept 4. The study of both equations and inequalities 5. Comparisons of algebraic and geometric representations 6. Leading the student to discover mathematios on his own. The subject matter of elementar,y algebra is essentially unchanged. The changes being recommended by various groups as the Commission on Mathematics and the School Mathematics Study Group are the stress of the development of concepts and meanings as being equal to mechanical manipula tiona. There haa also been a change with regard to vocabulary and symbolism. The concept of set is used early in ,the course and becomes a unifying and clarifying concept t~xoughout the course.' Inequalities are developed along with equations and graphs are used much more extensively. Geometr.y is undergoing a relatively rapid evolution. Geometry is being treated in conjunction with plane geometr.y instead of as a seperate course. 66 Trigonometry is disappearing as a seperate course and appears in a senior mathematics course along with vectors, exponential functions, Euler's formula, permutations and combinations, and Cartesian coordinates. School enrollment ~ Mathematics ~ssea. The percentage of students enrolled in mathematics classes as shown by the reporting schools was ve~J good. The 27.6 pe~cent enrollment in algebra followed by the 21.9 percent in geometry tme comparable with the national average. The drop to 11.45 percent of the student enrolled in algebra II and the fact that 38 of the reporting schools do not offer algebra II was a factor to consider. Also, 113 of the 145 reporting schools did not offer a senior mathematics program and S5 of the schools did not l~ve trigonometry and solid geometry classes. Small enrollment, and only one mathematics teacher employed were some of the main reasons why more schools terminate their mathematics program with geometry or algebra II. Mathematics ~~ noncollege £~pabilitl. There was a growing concern over the students who enroll in general mathematics. This course was not a terminal one for these students as two years 69 of mathematics 1s required in the secondary sohools and much of the material content of the course was of a 80cial nature. These two factors made this course a subject of discussion and revision by mathematicians. One of the problems under discussion was the grade level at which this course should 00 offered. The teacher opinions on this topic were varied. The grade levels suggested ranged all the way from the 8th grade to the 12th grade as illustrated in Table II, page 54. Eleven teachers said that the course should be dropped from the curriculum. The majority of the teachers reporting (12 teaohers or 49 percent) said that general mathematics should be taught in the 9th grade. A number of teachers (11 or 1.4 peroent) said the 11th grade would be the appropriate level. However, 29 teachers or 20.0 percent indicated that the 12th grade would be the most appropriate. The suggested revision of the general mathematics course indicated that the 11th and 12th grades might be the best grade level to teach the social topics of this course. This was the time when the students would be most interested in such subjects as budgets, taxes, checking accounts, and interest. .T...e...x..t..b. ooks used. The type of text ueed and how often old texts were replaced was of vi tr'll importance when revising a mathematics program. Most of the schools reporting ordered new texts every 3 to 5 years. The five year program was the most popular, with $, schools ordering textbooks by this plan. The tour year plan was next with 34 schools using this plan. There was some doubt in a few schools which seemed to have no set schedule of ordering new texts, and 20 schools reported that they order as needed. There was plenty of opportunlty to acquaint teaohers and school administrators with neu texts, as 117 of the schools indicat6cl that they would be ordering 70 new algebra and algebra II textbooks aft~r the 1963 school year. Th6 textbooks authored by rlelchons and Krickenbcrger were the most widely used by the Montana schools, with 29 percent of the schools using this text in algebra, 46 percent using it in plane geometry, 37 percent in algebra II, and !~9 percent using the book in trigonometry and solid geometry. The schoole whioh reported having a senior mathematics cla.ss used the textbook Foundations 2£ ~dvanced Mathematics by KlineOsterleW11eon in 42 percent of tlw classes; a 27 percent rise above any other book. CHAPTER V MATHEMATICAL TOPICS There has been increased emphasis in the mathematics currioulum in regardsl 1. the underlying mathematical assumptions 2. precise defini tiona and vocabulary 3. sets of elements as a unifying concept 4. study of both equations and inequalities 5. comparisons of algebraic and geometric representations.1 These emphases require a new point of view for teachers, students and textbooks. Background. This chapter dealt with Bome of the mathematical topics emphasized by the current thinking. Some ideas ot presentation of these topics was included to show bow the understanding of principles can relate to computation. These examples are not to be construed as advocating a single approac h to the study of' any topic. There must be flexl. bili ty which can be attained only through many approaches. Applications are a source of interest and motivation for pupils and can aid in olarifying mathematical ideas. lBruce E. Meserve and Max A. Sobel" Mathematics for Secondar,l School Teachers (Englewood CliffsJ PrenticeHaUt Inc. 1962), p. 7. 72 Table V, page 131 showed the distribution of the number of teachers reporting the difficulties of mathematical topics for the students. Mathematical vocabulary. Much attention was being given to make sure that mathematical statements are correctly verbalized, and that words used in presenting mathematical concepts are accurate and precise. For example. some texts make the failing in that a.n informal statement which a teacher might employ orally in making an offthecuff explanation is suitable to be printed as a rule to be followedl ftTo add like fractions, or fractions with the same denominators, add the numerators." This statement taken literally, would.mean that ~ + ~ • 7 Great care should be taken to make the mathematics correct, and not to present erroneous material that must be unlearned later. Ever.y conscientious teacher should make an effort to present what can be understood b.1 his students, what will be the most useful to them, and what he believes is true. The mathematical vocabulary used by the teacher is of the utmost importance as the student must be able to use and retain this vocabulary as he progresses in mathematios. The possession of an adequate set or mental symbols corresponding to mathematical relationships must be acquired by the student. 73 TABLE V DEGREE OF DIFFICULTY OF MATHEMATICAL TOPICS FOR STUDENTS AS INDICATED BY TEACHERS OF MATHF.HATICS Degree of Difficulty Percentage By Number of Teachers Most Mathematical Topic Diffioult Most Average Easy Mathematical Vocabulary 21 65 79 14.5 Signed Numbers 16 66 63 li.O Fractions ,1 48 40 39.5 Geometric Concepts 26 83 )6 18.0 Solving Equations 9 74 62 6.2 Evaluating Formulae 31 54 60 21.4 Verbal Problems 117 18 10 81.0 Graphing 19 51 69 13.0 Solving Systems ot Equations 18 ,9 68 12.4 Fundamental Operations Addition, etc. 4 70 71 2.8 Sete" 10 21 40 13.8 Series* 23 44 69 16.0 Neatness ot Papers 9 76 60 6.2 Checking their Work 21 S2 72 14.S NOTE I if6ets and Series not taught in all schbols. This table should be read as followal Under the mathematical topio fractions, 57 teachers repor·ted that students had much dif:ficul ty with fractions, 48 teachers reported students had an average amount of trouble with fractions, and 40 teachers reported the topic was easy for the students. or the total number of teachers. 39., per cent stated that the topic was most difficult for students to master. . The reporting teachers indicated that students had a comparatively easy time assimilating mathematical vocabulary. 74 or the reporting teachers, 21 teachers reported students had difficulty with mathematioal vocabulary, 6, said that students had little difficulty understanding and using mathematical vocabular,y, and 79 teachers reported that mathematical vocabular,y was easily picked up by the students. Signed numbers. Whenever there has seemed to be a good reason to do so, mathematicians have invented new sets ot "numbers". When one auch invention is made, it i8 not hard to realize that there is no reason to stop inventing. It is easy to invent things that do no work, but hard to invent things that do work and are useful as well. As more and more sophisticated mathematics were studied, there has been occasion to use more and more sophisticated kinds of Rnumbers". In arithmetic~ subtractions like 5  8 were impossible under the system of whole numbers and fractions. To remove this limitation on subtraction, new nun!bers were created. Thus: for 5  8 .. 3 was created so the S • 8 + (3). For these new numbers, there must be names for them, symbols for them, definitions of addition and multiplication of them, and proofs that addition and multiplication of them have the usual characteristics symbolic of ad~ tion and multiplication. One of the basic defini tiona for signed numbers is, "That for every + n, a number 75 n 18 erea ted corresponding to +n by the defini ti on (+n) + (n) • 0." Thus J n is to be read nega ti ve n and is also called the addi'ti ve inverse or +n because (+n) + (n) • O. Also the number n from which +n and n are created, is the absolute value of both +n and n. While the concepts of signed numbers seem comparatively easy tor students to grasp, (only 16 ot the reporting teachers indicated that this topic was hard for students to master) students tend to neglect these signs when engaged in computational skilla. Clear understanding of the concepts ot signed numbers will help the student think through more complex mathematical situations without being cluttered mentally by errors as to the positiveness or negatlveness o~ a number. Fractions, In the system or whole numbers J there i8 no quotient for such an indicated division as 9 • 5. To remove this limitation, new numbers called fraotions were created. A fraction is an ordered pair ot whole numbers J as the fraotion 6 where b i8 not zero. The definition, i · a, makes every whole number a member ot the set of fractions. Students have little trouble multiplying tractions as: ~ • a. + sa· However, making students .see this operation in reverse i8 sometimes very difticul t. Work with fractions are built out of simple fractions and must be broken down in order to be 801 ved. If students receive practice in using the properties 76 of 1 as illustrated by the definitions _a . 1, and breaking a fractions down by reversing the multiplication process, their work in solving complex fractions would become easier. Fractions was one of the topics teachers reported students had the most difficulty with. or the teachers reporting, 57 or 39., percent reported that students experienced much difficulty in manipulation ot fractions. Forty teachers reported that fractions were easy for the students to become proficient in and 48 reported that students experienced a normal amount of difficulty in working with fraotions. Geometric concepts. One of the major concerns in the introduction of geometric concepts is careful attention to definitions. As in any logioal system, it is not possible to define everything~ Usually a point and a line are taken as undefined relations. The property of a point being on the line or plane is also taken as an undefined relation~ Then careful definitions are drawn in terms of these assumed terms and relations. In geometry, there is a necessity for precise definitions, emphasis on the nature or proof, including the significance of assumptions (postula tee and axioms). The following concepts or topics appeared in most of the newer textbooks though not necessarily as they are listed here. 1. Defini tion and history of geometry 2. Nature of a proof 3. Measurements and constructions 4. Triangles 5. Parallels 6. Inequalities, coordinates, and loci 7. Polygons 8 •. Circles 9. Areas and volumes 10. Numerical trigonometr,y ll. Deductive and induct! ve reasoning 77 Teacher opi.nion on the difficulty of each of these ooncepts was limited by the length of the questionnaire, and the fact that some of these concepts had not been reached at the time the questionnaire was sent out.· It has been the experience of the writer that student achievement in geometric concepts depends chiefly upon how well the student grasps the definitions and terminology relating to the unit. Geometric concepts were reported as difficult for the students by twentysix of the reporting teachers. Normal student difficulty was reported by eightythree teachers, and thirtysix teachers reported that geometric concepts were easy for the students to assimilate. Solving equations. When a statement is in the torm or an equation, the statement 18 interpreted to mean that the expressions on either side of the symbol, ., mean or stand for the same thing. Recently it haa been proposed that equa tiona likel ~ • 6 be called "open sentences". This was done to emphasize that, as a sentence, it is not true fot' all values of YJ that it is true for only one value and even then it may not be one of the permissable values under consideration. Solving equations was a comparatively easy concept for students to handle. Only nine of the reporting teachers indicated that this concept was very difficult for the students 78 to grasp. That the concept of solving equations was put across to the students with normal amount of effort, was reported by seventyfour teachers. Sixtytwo of the teachers reported that solving equations was an easy topic for the students to master. Evaluating Formulas. Formulas were the most useful part ot algebra to the vast majority ot people. In the presentation ot how to evaluate formulas, two alternatives present themselves. One altern~tive i8 to assume t~at the formulas that are understood most easily by students are ~hose related to geometrical figures. It appears that the C9mmission on l!athematics Ir~de the assumption that all students are familiar with geometrical figures and their formulas. 2 2Walter W. Hart and others, New First Year Algebra (Boston: D. C. Heath and Company, 1962), p. ro: The other 18 that students have a great deal or trouble mastering percentage formulae.· 79 Thougb solving equations and evaluating formulas are quite similar in mechanics of computation. thirtyone of the reporting teachers stated that students had much difficulty with formulas. This was a 15.2 percent increase over the teachers that reported much diffic\tlty in solving equations. Of the teachers responding, fiftytour reported that the students experienced a usual amount of difficulty and sixty teachers reported that evaluating formulas wae an easy concept for the students to grasp. Verbal problems. One of the most difficult parts in the teaching of rna thema tics is teac bing students to translate the works ot a verbal problem into mathematical symbols. Many devices have been reco~ended, many nspecial tricks" have been devised. The student gets the impression from these tricks and devises that there is no general approach to the solution ot verbal problems. He feels that with each type of verbal problem there 18 a particular trick or devise needed to solve the problem. The student i8 often required to transla.te verbal problems without having been well Tersed in mathematical vocabulary and t~ relationship between mathematical 6,y.mbols and words. To help students solve problems, moat text books still use a standard procedure. A type problem is given, explained by an illustrative example (usually in different colors), and followed 80 by problems involving the same teohniques for solution. Thus the student memorizes a procedure .and develops a skill, but beyond reoognition of a type of problem, does no real thinking. That students are still struggling to solve verbal problems is evidenced by the fact that 111 of the reporting teachers stated that students had most difficulty with this mathematical concept. Only 18 teachers indicated that students grasped this concept with normal effort and 10 stated that students had an easy time with verbal problems. Graphing. Graphs as an aid are at once a means of clarifying definitions, and another means of defining mathematical relations. Graphs are effective ways of presenting mathematical functions. A start is made by plotting funotions which are sets of ordered pairs of whole numbers resulting in a dot graph. Line graphs of functions may be developed later and used in the solution or systems of linear equations and in the study of inequalities. Graphical representation i8 employed to ,aid in £urniuhing the necessary instruction of systems of quadratic equations, and graphs of seta of numbers on a line. Students seem more able to express their thinking in ter.m of graphs than by translating verbal problems. Of the teaohers reporting, 19 indicated that graphing gave the students quite a bit of difficulty. Many of the teachers felt that graphing was a comparatively easy concept to grasp as fiftyseven 81 teachers reported presenting graphing to the students with a normal amount of effort, and sixtynine teachers saId that graphing was an easy concept for the students in which to achieve proficiency. Solving systems £! e~at1ons. Instruction about the solution of systems of equationsone of which is the systems of quadratic equationshaa been simplified during past years. Quadratic equations are first solved graphically to clarify the fact that there may be more than one real root and to prepare for subsequent illustrations when there might not be real roots. The procedure in drawing such a graph of quadratic equations assumes that the graph is a smooth andcontinuous curve. Though solving systems of equations is a more sophisticated procedure than solving a linear one, students still do not experience much difficulty in achieving in this ooncept. Only 18 of the teachers reported that students had an exceptional amount of difficulty with this concept. That students had to put forth a normal a~ount of effort to learn this concept was reported by 59 teachers, and 68 reported that the concept of solving systems of equations was easy for the students. Fundamental operatio~. MUch of the mect~n1cal operations of secondar,y mathematics depends upon the fQur fundamental operations of addition, ~\btraction, multiplication, and division. 82 Addi tion is in! tially defined in terms of counting and is an operation which assigns to a pair of numbers another number called the sum. Subtraction, on the other hand, may be defined in terms of addition; it is the operation for finding one addend if the sum and the other addend are known. The fact that Bubtraction is the inverse operation ot addition should not be obsoured. The conoept of roul tiplication has usuallybeen presented as repeated addition. While it is not suggested that this approach be abandoned, it could be Bupplemented by the cartesianproduct interpretation. Division should be defined in terms of multiplication just as subtraction is defined in terms of addition. Facility in computation cannot be separated from the significant contribution wAde to it by an understanding of the operations already emphasized. The pupil should aohieve a degree of maste:;:y of these operations which will enable him to think through mathematical situations without being cluttered mentally by errors. Students perform matherr~tical operations only to find that they have made an error in performing one of the fundamental operations which nlllified their result. Only four teachers reported that students had much difficulty with fundamental operations. The rest of the reporting teachers were evenly split in opinion with 70 teachers stating that students had the usual amount of difficulty, and 11 teachers stating that the fundamental operations were easy for the students. 83 ~. The concept of set 18 one of the major items being recommended for inclusion in the secondar,r school mathematios curriculum. The concept of a set is fundamental for communicating ideas in mathematics, just as it is in everyday language. Groups, herds, organizations, and teams are oollections of elements which comprise sets. Mathematics textbooks could be written without using the word set, but not without using the concept. It is claimed that this notion can be used as a unifying and clarifying concept throughout mathematics in the secondary school curriculum. The concept of a set is taken as undefined in mathematics. Sets should be thought of intuitively as collections of objects. It is possible t~ look at an object and tell whether it belongs in a set. No mention of sets should be made unless an effeotive use is made of the ~rminology and concepts in subsequent mathematical development. Not all the secondal" schools include sets in the mathematical topics considered. Of the 71 teachers reporting, 10 said that sets was a difficult topic for the students; twentyon~ said that students experienced a normal amount or difficulty, and forty said that the concepts of a set were easy for the student to understand and use. 84 Series. The concept of series was taken as the topics or sequences and progressions that appear in algebra II. A sequence of numbers is an ordered eet of numbers. It ~~plies that the numbers are written in order so that there is a first one, and each has a successor. A progression is a Bequer~e of numbers each of which, after the first one, can be obtained by combining a constant to the preceding. Series have more immediate interest for the stUdents than much of the theoretioal mathematics, because they relate to problems of ia~ediste concern to people. Also, the definition 15 ~~ed in with the subject of sets of numbers, a conformity recommended by the Co~~s8ion on Mathematics. Much of the work with series consists of mechanical manipula.tions and use of formulas. Again, according to the teachers reporting, the students do not encounter much difficulty with the mechanical operations. or the 136 teachers reporting, twentythree stated that the concept of series was difficult; fortyfour indicated that students grasped the prinoiples involved with a normal amount of effort) and sixtynine teachers reported that these concepts were easily grasped by the students. Neatness .2! papers. Students can rectify many of their errors if they can look over their papers without being confronted 8, by a mass of symbols ,and figures scattered over the paper without any apparent organization or arrangement. Teachers will also get a better idea of the pupills work if they can follow ·~he mathematioal operations from start to finish with a clear understanding of what the student is attempting to do. \ihen the paper is returned to the student, he cnn readily assertain where his mistakes occl1.red and how he can avoid them in the future, if the paper has been done in a neat and orderly manner. Montana high school students seemed to have achiG~ed some degree of success in this ~roc€ss as only 9 teachers reported difficulty with students turning in untidy papero. That students have some dlfficulty in keeping their papers presentable, was reported by seventysix teachers, and sixty teachers stated that there is no difficulty encountered with students being lax about the neatness of their rr:athematics papers. Ohecking~. Accuracy is one of the prime requisites of mathematics. Time and time again, students perfor.m all the manipulative operations of a problem" but end up with an incorrect solution because of carelessness. Much of this carelessness can be eliudnated if students will truce a little time and check their work over before handing it in. As students check over their work, they ~~ll become conscious of where they are making mistakes and tend to gain confidence in their work. 86 The students tend to have more difficulty with checking their work than with neatness. or the teachers reporting, twentyone indioated that students had much difficulty with checking their workJ fiftytwo teachers stated that the students eXi1erienced a normal amount of trouble in remembering to oheck their work; and seventytwo teaohers reported that the students were careful about checking their papers. Summary. There has been remarkable advances in the clarity and precision of mathematical discourse. PAny of the apparently diverse notions are special cases of a few underlying concepts. The conoepts presented in this chapter were an overview of topics presented throughout the secondar,y mathematics curriculum. Because sound and meaningful concepts can be just as poorly taught 8S unsound concepts J some of ·~he ourrent thinking as to the presentation of this topic in the classroom was included. A point of controversy in this chapter was what constitutes the dividing line between much difficulty, a normal amount ot effort on the part of the student, and what is easy for the student to grasp. It has been the experience of the writer that whereas the amount of difficulty of a topic for a student may be difficult to explain, a teacher can tell intuitively whether or not a topic is more difficult for the student than another topic 87 This chapter again brought out the fact that students have little difficulty doing mechanical operations as evidenced by the fact that 6.2 percent of the teachers reported much difficulty on the student's part in solving equations, and 12.4 percent in Bolving systenls of equatlvns. However, 81.0 percent of the teachers reported much student difficulty in doing verbal problems. CHAPTER VI TEACHER OPINION ON THE MATHEMATICS CURRICULUM In a study of this nature, it was neceesar,y to get teaoher opinion on what could be done to improve the mathematical structure, and their criticism as to what is wrong with the mathematical structure in Montana schools. Background. Not all teachers put down opinions and some opinions consisted of only a word or two. Because of the fact that some teachers did not want their names mentioned in oonnection with this stu~J the number of comments that could be inoluded was limited. A list of the teachers who contributed was included in the preface. The opinions were not taken verbatim but oare was taken to insure that the original meanings were not changed. The opinions were d1 vided into three groups t (1) opinions on curl·iculum, (2) comments on the elementary school program, and (3) teacher orientation. The first two groups constituted this chapter. The third group 18 contained in Chapter VII which dealt with teacher training. 89 Teacher opinions .2!l curriculum. Modern mathematics needs to be defined. Goals should be set up, and all teaohers should know what the goal are and his or her expected contribution. It does no good for a 5th grade teacher to attempt to introd~ce algebra if the 6th grade teacher is going to ignore algebra. A coordinated program is a must. Also, extension college oourses on changing mathematical needs would be a great help, as many teaohers need refresher courses in mathematics. The concepts in the grade school must be updated so that use can be made of these conoepts in the high school, instead of trying to do both at the same time. This updating should be done slowly and in coordination with all mathematics teachers from grade one to grade twelve. This would enable one hand to know what the other is doing. For this to happen, the school boards and administrators must be educated to see the advantages of change, and provide time in which the teaohers could work to make this change possible. It will be a great benefit when a tested course of study evolves out of the pres~nt revolution in mathematics. This will take time, but a course of study 1s needed especially for the senior year. The area of verbal problems should be improved becauBe this is how the students will really use their mathematics. Though fractions in the grades could be improved, the philosopqy of pushing this modern oonoept of mathematics into the lower grades is not praotical. Teaoh the students fundamentals down there. A few will be able to grasp the modern concepts, but the rest will be so confused that they will not be able to add their own grocery bill. The modern trends in mathematics should be kept in the junior high school and high school. 90 There haa been too much improper orientation of students for future work. Improvement could be made in the classification of the students for enrollment in mathematics courses. The students with poor backgrounds should be enrolled in general mathematics or remedial arithmetic; those with higher aptitudes should be placed in accelerated oourses. Most groups, espeoiallyin small sohools devote very little time to the courses of mathematics. Courses like algebra II or trigonometry are not taught at all or are taught on an alternate year basis. In some of these small schools, there is no teaoher with a mathematics major teaching mathematics. Schools have not gone to the modern mathematics program because they are not sure how to turn. Small 8chools would like to change their curriculum, but just do not knoW' how. They cannot offer tour years of mathematics straight through because ot the small student enrollment. How can some advanced concepts be taught without having algebra II first? To teach the new concepts, sets make mathematics understandable and explain wqy, instead of a lot of rules and do this or that. 91 The teacher should make sets serve as a tool of understanding. Reading and interpreting what is read plays a major part in the student's success in mathematics. Improvement should be made in the areas of reading and logic. Lack of agreement in terminology, and when to' present certain materials, among the various authors, leads to some of this hinderence. Also~ the influx or college material to the high schools before the students are rea~ to handle it, is a cause of concern. The ability to read and comprehend verbal problems is one of the major weaknesses in the ~Bthematics ability of the student. If the students could improve their ability to read, it would solve most or the problems the student has as far as mathematics is concerned. Basic fundamental operations and the translating of verbal situations into workable mathematics forms need improving. Students are slow in assimilating and correlating the total mathematics pictUl~. Vocabular,y should be stressed along with the basic operations, and the "algebraic laws" which are the basis or mathematics in school. The textbooks should emphasize the vocabular,y of mathematics and a glossary of mathematical terms and principles should be included in every textbook. Oral exercises, starting with Simple problems that demonstrate new 92 principles and progress to more difficult ones, is also needed. Most of the students have very little reaeoning power. They know most ot the "hows" but are unable to work and think in an organized manner. They, in general, 'are lacking in the very basic ideas of mathematics. "I want the answerto heck with the method." Inadequate preparation in the comprehension of written problems 1s a difficult one to correct when so many students take only two years of high school mathematics. Generally, students seem to have a good foundation in essential arithmetic processes. Modern mathematics in the high school curriculum can be appreciated only by the better students. The content and apprach of some of the newer textbooks as the School Mathematics study Group., Ball State series and others is highly approved. However, the biggest need is to change the attitudes of the teachers, students, and parents to the newer approaches to mathematios. More work needs to be done in mathematics courses for slower learners. In classes for average or below average students, the value ot traditional mathematics seems dubious, but answers to just why this 1s, are difficult to assemble. Perhaps algebraic and geometriC concepts should be introduced throughout the grade school.' Primary children seem more willing and able to grasp ideas better than high school students •. 93 Some of the valuabl~ traditional mathematics is becoming neglected tor more "sophisticated" mathematics. The mathematics curriculum oertainly needed to be revised, but for the nonmathematics major, the old algebra and trigonometry is possibly more useful. The new programs are wonderful for the exceptional student. General mathematios should be improved. Some small schools would like to revise the mathematios ourriculum but as in a case like ours, the textbooks are not worn ao~pletely away, so new texts cannot be ordered. The school board is extremely conservative~ No National Defense Education Act is allowed in the sohool. The facilities are extremely poor and no money will be spent on improvements. Finances are no worry as this is one of the more weal thy districts. General mathematics or consumer mathematics should be a more practioal course rather than a dimping ground for those students who are unable to pass algebra and geometr.y. The area of mathematics for the noncollege bound lacks textbooks and currioula context. More practical problems should be presented in the development of algebraio concepts. The modern mathematics program is a "must" for the future. Geometry is much too "old fashioned", and much of it could be either simplified or dropped from the curriculum. Textbooks at the high school level should be published 94 in the areas of logic, sets, analytical geometry, and other topics. So far, the textbooks being used are not strong in the development of the number system, the language, and use of sets. More mathematics can and should be taught at the elementary level. Most teachers have little more than six to eight hours in mathematics and few if any understand the new vocabulary. Institutes also spend too much time trying to teach over the head of the teachernot teaching the teachers anything. Hore emphasis should be placed on space relationships. Students also have difficulty working with inequalities. Verbal problems also pose a problem for most students. Most of this dirricul ty stems from the inability of the student to read and determine ~mat is being asked in the p~oblems. Geometry is in need of great improvement. The students find it difficult to comprehend the theorems and problems. There seems no way to improve instruction and make it easier for the students. There should be a nelT approach to geometry, providing intereet and stimulation to the students. Basic understanding of the various opera tiona tha t must be used in mathematics and how they all relate to each other in problem solving.. Topics such as integral domain, rings, fields, groups and set ideas are presented to the student every BO often to give the students an idea of what they will encounter in the field of mathematics.. The students bas1cal~ understand 9$ these topics but connot see any concrete use for them. They then lose interest and fail to understand many of the concepts. Theor,y in mathematics needs improvement. Modern algebra concepts are included in algebra II and the student cannot seem to get enough of it. They thoroughly enjoy advanced algebra. 9p:i.nions_ .2E. ~ elementary school 1?rogram. The primary grades is where the basis of the "modern approach" should be laid. The whole underlying foundation of our elemtntary mathematics program must be r€vised with much effort pu~ forth to replace the image of the traditional with focus on the new. E1ementa~J schools do not teach the kind of mathematics that is needed for success at the secondary level. The students don't know the "why" they do something, only the "hewn. The concepts of modern mathematics schould be introduced at the primary level and should be intensified in the junior high school. The more able students should be channeled toward the abstract concepts. The freshmen are not equipped to handle high school mathematics as a lmole. They are not capable of doing the simple operations as readily and as easily as they should be lfathema.tios needs its biggest boost from reading in the lower grades. The students have little difficulty with the major portion of the problems presented but 85 to 90 percent of them have trouble wi th verbal problems. This is shown throughout the state of Montana and the starting point for improvement in this topic is in the primary grades. More concepts should be taught in the grades besides just arithmetic. It seems that very little material in the field of geometr,y is taught in the grades. Enough algebra should be taught in the grades so that freshmen could take algebra II. This would help the students in their science courses also. The most improvement in mathematics needs to come in the elementary and junior high areas of mathematics. The junior high school students shottld either be given a course in algebra or one in insurance, taxes, banking, and other social topics. This integrated course is duplicated in the bigh school general mathematics course or in the consumer mathematics course, and is a waste of time for the better students as well as the poorer ones. Since it 18 difficult to allot time £or teaohing 96 basic arithmetical processes to high school students, a big step forward would be to improve the ari t hme tic and mathematics programs in the elementary school.. Some elementary teachers tend to skip over arithmetic or teach it in a fashion that doesn't do it justice. They tend to indoctrinate the students on uhow hard mathematics will be for them". Arithmetic and 97 mathematics at the elementary level sbould be taught in such a manner that the transition to high school and ultimately college mathematics would seem to be a natural step upward. Instead, to 9th grade algebra students, the laws of algebra seem to contradict what was taught to the students in elementary school. For example, to subtract signed numbers, the sign of the subtrahend is changed and the numbers added. Even when addition is performed, subtraction takes place if the eigne are unlike. This difficulty of negative numbers could be avoided if they were introduced in the 4th, 5th, and 6th grades. There is no reason why negative numbers cannot be understood if positive numbers are. The mathematics program in the elementary school should be revised and the mathematics program should adapt to change just as is done in any other curriculum area. However, we shouldn't depart from proven good concepts of content and methods of teaohing just for the sake of change. More rigid training should be provided in the elementary school so that students will have mastered the fundamental operations that cause most of the errors made by the students in high school mathematics. The elementary students need work with basic operations. This would enable the teacher of high school mathematics to get right to the subject matter at hand rather than spend more time on fundamental operations. This should b& accomplished in the student's grade school years. 98 The most improvement in mathematical areas should come before the student gets to high school and should come especially in the junior high school. The freshmen taking algebra just haven't had enough hard 'YTork to be prepared for algebra. Just to mention one thingfractions should be taugbb to such an extent in the grades that they would not have to be completely retaught in high school. Summary. Modern mathematics needed to be defined and goals set up so that all teachers would know what was being strived tor and what was expected of them. A seasoned course of study would help the small schools decide on how to approach the problem of devising a course ot action in mathematics. Reading and interpreting what is read plays an important part in the student's success in ma~hematics. Lack of agreement in terminology and the sequence of presentation of certain topics has contributed to a lack ot comprehension on the part of the student. It the students could improve their ability to interpret words into mathematical symbols J much of the difficul ty enoountered by the student would have disappeared. General mathematics should have been improved and not be a dumping ground for students who cannt achieve in algebra or geometry. 99 Students are interested in abstract topics in mathematics such as vectors, rings, and integral domain, but quickly lose their interest when they cannot see any conorete use for them. The elementar,y mathematics program needs revision as this is the starting plaoe for the concepts of mathematics. Elementary teachers tend to skip over arithmetic or teach it in a lackadaeical way. They tend to indootrinate the pupils on ~how hard mathematics will be for them". The fundamental operations of arithmetic should have been sufficiently presented so that they will not have to be retaught when the student reaches high school. CHAPTER VII TEACHER ORIENTATION The reorganization of mathematics in the secondar.y school posed a problem for the orientation of mathematios teachers. This orientation must include three areas of stu~: (1) modern rr~thematicsset theory, logic, and topology; (2) the need of a mathematics course for the noncollege bound student, and emphasis on the psychology of teaching this student; and (3) the procedures for the revision and the inlplementation of programs in mathematics education to meet the changing needs of society.l Background. The previous chapter mentioned a need for revision of the elementary mathematics curriculum, and a need for teacher orientation in the newer mathematical concepts. What is being done to help the elementary teacher was the main topic of this chapter. vlide reading and research seems to indicate that the elementary teacher i8 the one who is the least trained in mathematics and yet bears the brunt of the criticism for student failure in high school; still little is done to aid the elementar.y teacher in mathematics. Table Vll, page 102, showed the number of schools having lHoward F. Fehr, op. cit., p. 31. 101 Bome type of elementary teacher orientation. Teacher opinions, and a program the writer used in presenting the newer ~Bthematical ideas to the elementary achool teachers in the Columbia School District, Burbank, Washington, comprised the remainder of the chapter. Teacher opinion. If the "newU textbooks do the job expected ot them, then the area for improvement lies in the training of the teachers. Too many children are being taught that mathematics is hard. This could be prevented if the elementary teachers were trained and competent in mathematics. There are many and varied opinions about the revision in mathematics'. The textbooks are beginning to catch up, and now the problem is to get teachers that can competently teach what is presented in the textbook. Since the elementary teachers usually are the ones with the least training in mathematics, effort should be centered in that area first. There should be some organized method of making programs of teaching the new concepts of ~Athematics available to the elementary and secondary teachers in schools of small enrollment. ~ inservice course ~ elementary teachers. More and more appropriate courses in elementar,y mathematics are offered at Hontana State College and Ilontana State Uni versi ty • Teachers = Number of Schools 24 21 13 2 Total 60 TABLE VI SCHOOLS HAVING ORIENTATION PROGRA¥~ FOR ELEMENTARY TFACHP.~ ; Type of Program Inservice Institute Both Inservice and Institute other type ot Program 102 NOTE, 8S schools do not have elementary teacher orientation programs to acquaint the teachers with the new concepts and methods of teaching mathematics. 10) should attend these oourses whenever feasible. Since summer sessions are not always possible for teachers to attend, inservice training "classes" have proven to be extremely valuable and well received. The following program was used by the writer to orientate the teachers of the Columbia School District, Burbank, Washington, during the 196263 school year. The program was derived from suggestions received at the Northwest Council of Teachers of Mathematics at Gearhart, Oregon) an institute on elementary school mathematics at the University of Washington, Roy Dubisch, director) and from the inservice program presented to elementar,y teachers in the Seattle Public Schools. The program was well received by the teachers and helped the elementary teachers considerably in assimilating new mathematical concepts. MATHEi1ATICS FOR ELEHENTARY TEACHERS There has been more mathematics discovered in the past 50 years than in all time previous. People must know more in order to function effectively in the complex society of today. Are the grade school students of today going to be prepared and have sense enough to operate the society they will inherit? Our task is to provide every pupil today with the mathematical instruction which will be m.ost useful to him 104 tomorrow and make him most useful to society. 'Ever,y child should at least be given the opportunity to learn the mathematics which will enable him to achieve in a highly technological society. In the past, mathematicians have had a very lax attitude toward exposition. The readability of mathematical work was confusing, and too much use was made of such phrases as nit is obvious that", "thuB it is readily seen u, and tt1 t clearly follows". It was as though the mathematicians would be a li ttle crushed if they were perfectly understood~ There have been remarkable advances in the clarity and precision of mathematical discourse. The concept of sets is aimed at clarifying many mathematical terms that have been rather vague to the student. A sound ourriculum 1s not sufficient. Sound and meaningful concepts can be just as poorly taught as unsound concepts. A reorientation ot the mathematical program which emphasizes structural aspeots will be unseccessful unless the pedagogy is successful. Good mathematical instruction has a dynamic charaoter. Pupils should be encouraged to make conjectures and guesses, to experiment and formulate, and to understand. The teacher should make the student think and make him uncomfort 



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