mployment in Levº Professional Mathematical Work in Industry and Government NATIONAL SCIENCE FOUNDATION NSF-62–12 UNIVERSITY of MicrºCA, GENERAL LIBRARY Employment in Professional Mathematical Work in Industry and Government Report on a 1960 Survey Prepared for the National Science Foundation by the U.S. Department of Labor Q ( →. Bureau of Labor Statistics, in cooperation with ,” The Mathematical Association of America NSF 62–12 CA \\ .5 .U. S.44 / ?4 2– For sale by the Superintendent of Documents, U.S. Government Printing office Washington 25, D.C. - Price 55 cents wº-yº”T A. erººc/ TED BY T unree sºrts 3. Sºnic, (2-/4–6, 2. ACKNOWLEDGMENTS This study of employment in professional mathematical work in industry and Government was carried out under the sponsorship of the National Science Foundation, Division of Scien- tific Personnel and Education. It is one of a series undertaken in that Division's Scientific Personnel and Education Studies Section as a part of its function of maintaining a “clearing- house for information covering all scientific and technical personnel in the United States. . . .” Thomas J. Mills, Section Head, and Robert W. Cain, Program Director, Scientific Manpower Studies, provided overall guidance and assistance. The survey was planned and conducted and the report was prepared in the U.S. Depart- ment of Labor's Bureau of Labor Statistics. The planning was under the supervision of Helen Wood. The survey was conducted and the report was prepared under the direction of Cora E. Taylor. The tabulation plan was devised, data were analyzed, and the report was prepared by Harry Greenspan, with the assistance of George A. Hermanson and Jesse L. Davis. Thomas F. Mosimann designed the questionnaire; Hyman B. Kaitz acted as statistical consultant; Brendan J. Powers devised the data-processing plans; and Edmund W. FitzGerald directed the machine tabulation operations. Throughout the planning and preparation of the report, the Bureau of Labor Statistics was guided by an Advisory Committee of the Mathematical Association of America, which was chaired by Dr. Morris Ostrofsky. The National Science Foundation and the Bureau of Labor Statistics wish to express their appreciation to the companies and organizations whose cooperation made the project possible and to the individuals who supplied the data on their employment. PREFACE This reportſpresents the findings of a survey of mathematical employment other than teaching. The survey, conducted by the Bureau of Labor Statistics for the National Science Foundation and the Mathematical Association of America, was undertaken to provide current information on the Nation’s manpower resources in the rapidly growing field concerned with applications of mathematics. The detailed information on mathematics courses required for work in this field and on other characteristics of mathematical employment was assembled to aid educators in planning the revision of mathematics curriculums to meet the changing requirements of the Nation’s technology. - To locate individuals, other than teachers, whose positions required college-level mathe- matics training, questionnaires were sent to a sample of companies and to Federal Government agencies and private nonprofit organizations known to employ mathematicians. The criteria given the employers for distribution of questionnaires to employees were broad enough to cover not only individuals with the title of mathematician but also persons with titles such as com- puter programer, operations research analyst, mathematical statistician, actuary, research engineer, and engineering analyst if they were professional personnel engaged in primarily mathematical work. Usable information was received from about 10,000 respondents. Data collected on the age, education, experience, and other characteristics of these persons, as well as on the nature of their current positions, functions performed, and income received, are discussed in this report. a Because of its special interests, the Mathematical Association of America plans to prepare a separate report based on this survey which will analyze the data on the mathematical content and educational requirements of positions held by respondents. The separate report will also discuss the implications of the survey findings for planning improvements in mathematical training. Preface CONTENTS * * = * * * * - - - - - - - - - - - - - - - - - - - - - - - - - sº - sº - - - - ess - sº º sºme s = ** = <= * = ** = ** = * * = ** = * * * * * = ** = * * *e = * * * * * * = * = * * * * * * * * * * Highlights------- — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — Characteristics of Personnel in Mathematical Employment------------------------------------------------- Age and Sex------------------------------------------------------------------------------------- Educational Attainment----------------------------------- ---------------------------------------- Major Subject Field of Degree--------------------------------------------------------------------- Extra College Credits Earned------------- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Age at Receipt of Degree------------- - — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — Timelapse Between Degrees----- * * * * * = = * * = <= ** = * = - * * * = ** * = * * * * = * * = ** = = * * * * * = * * * * * * * * * * * = ** = * * * * = * * * = * * * Professional Experience--------------------------------------------------------------------------- Professional Work Outside Major Position----------------------------------------------------------- Job Changing, 1950-00---------------------------------------------------------------------------- Characteristics of Mathematical Positions, 1960--------------------------------------------------------- Type of Employer------------------------------ * = as a , º, am - amº as * * * * * * = * * * * * * * = * = * = * = * * * * * * * = * = * * = * * = * * * * * Educational Level by Type of Employer-------------------------------------------------------- Size of Employing Company----------------------------------------------------------------------- Educational Level by Size of Company--------------------------------------------------------- Organizational Unit of Employment---------------------------------------------------------------- Computing Laboratories---------------------------------------------------------------------- Mathematics Units--------------------------------------------------------------------------- Operations Research Units-------------------------------------------------------------------- Research and Engineering Units--------------------------------------------------------------- Statistical Units------------------------------------------------------------------------------ Other Units--------------------------------------------------------------------------------- Operational Status------- ------------------------------------------------------------------------ Educational Level---------------------------------------------------------------------------- Organizational Unit-------------------------------------------------------------------------- Number of Employees Supervised.-------------------------------------------------------------- Mathematical Education Required for Current Position----------------------------------------------- Education Required Related to Attainment------------------------------------------------------ Income from Mathematical Employment, 1960----------------------------------------------------------- Educational Level, Age, and Sex------------------------------------------------------------------- Type of Employer-------------------------------------------------------------------------------- Supervisors and NonSupervisors-------------------------------------------------------------------- Additional Professional Income-------------------------------------------------------------------- A. Statistical Tables---------------------------------------------------------------------------------- B. Scope and Method-------------------------------------------------------------------------------- C. Questionnaires and Covering Letters----------------------------------------------------------------- Table Table Table Table Table Table : 6 TEXT TABLES . Persons in mathematical employment, by age group and Sex, 1960-------------------------------- . Age distribution of persons in mathematical employment, by broad industry group, 1960------------ Major subject field of highest degree held by persons in mathematical employment, 1960------------- . Credits earned beyond highest specified educational level, for persons in mathematical employment, 1960– . Distribution of persons in mathematical employment, by number of years between baccalaureate and advanced degrees, 1960-------------------------------------------------------------------- . Distribution of persons in 1960 field of mathematical employment by field of prior experience---------- 33 67 71 : TEXT TABLES-Continued Table 7. Percent of persons in mathematical employment who had professional work outside their major position, - by educational level and Source of Secondary employment, 1960--------------------------------- Table 8. Annual rate of job changes by persons employed in mathematical work, by educational level and experi- ence, 1950-00----------------------------------------------------------------------------- Table 9. Educational level of persons in mathematical employment, by employer, 1960---------------------- Table 10. Educational attainment of persons in mathematical employment in private industry, by size of company, 1900------------------------------------------------------------------------------------ Table 11. Mathematical employment, by type of organizational unit, 1960---------------------------------- Table 12. Operational status of persons in mathematical employment, by major field of education, 1960-------- Table 13. Operational status of persons in mathematical employment, by educational level, 1960--------------- Table 14. Comparison of educational level attained and level of mathematical education believed required by respondent, for persons in mathematical employment, 1960----------------------------------...-- Table 15. Extent of use of minimum mathematical education cited by respondents as needed to perform the duties of their position, 1960--------------------------------------------------------------------- Table 16. Major function of persons in mathematical employment, by type of employer, 1960----------------- Table 17. Distribution of persons in mathematical employment, by annual income from major position and by educational level, 1960--------------------------------------------------------------------- Table 18. Median annual incomes of men and women in mathematical employment, by educational level, 1960--- Table 19. Median annual income of persons in mathematical employment, by principal type of employer and - educational level, 1960--------------------------------------------------------------------- Table 20. Amount and percent by which median annual income of supervisors exceeded income of nonsupervisors in private industry and Government, by educational level and selected age groups, 1960- - - - - - - - - - - Table 21. Percent of persons in mathematical employment with income from professional work other than their major position, by amount of additional income at each educational level, 1960------------------- CHARTS Chart 1. Women were concentrated in the younger age groups-------------------------------------------- Chart 2. Proportionately, more men than women have advanced degrees----------------------------------- Chart 3. Advanced degrees are earned over a broad age range-------------------------------------------- Chart 4. Four out of ten employees had less than 6 years of professional experience------------------------- Chart 5. The majority have worked for no more than one employer--------------------------------------- Chart 6. Computing laboratories had the youngest group of mathematical workers; administration units the º oldest----------------------------------------------------------------------------------- Chart 7. Educational attainment by type of organizational unit, 1960------------------------------------- Chart 8. The majority of respondents 45–54 years of age were administrators or supervisors---------- — — — — — — — — — Chart 9. Within the same age group, approximately equal proportions at each educational level held administra- tive or supervisory positions---------------------------------------------------------------- Chart 10. Research was the chief function of mathematical workers in most fields of employment---------------- Chart 11. Median incomes were generally higher for persons with advanced degrees-------------------------- Chart 12. Incomes were highest in private industry------------------------------------------------------- APPENDIX TABLES Table A-1. Educational attainment of men and women in mathematical employment, 1960------------------ Table A-2. Age, sex, and educational attainment of persons in mathematical employment, 1960 - - ------------ Table A-3. Educational level and major subject field of persons in mathematical employment, 1960----------- Table A-4. Shifts in major subject field between baccalaureate and advanced degree, for persons in mathematical employment, 1960---------------------------------------------------------------------- Table A-5. Age at receipt of highest degree of persons in mathematical employment, 1960------------------- Table A-6. Age at receipt of bachelor’s degree of persons in mathematical employment, 1960----------------- Table A-7. Years of professional experience of persons in mathematical employment, 1960------------------- Table A-8. Number of employers in last 10 years for persons in mathematical employment, by experience level, 1900---------------------------------------------------------------------------------- Table A-9. Distribution of persons in mathematical employment, by age group, for each type of organizational unit, 1900----------------------------------------------------------------------------- Table A-10. Educational level by organizational unit for persons in mathematical employment, 1960----------- Table A-11. Operational status of persons in mathematical employment, by age and educational level, 1960----- Table A-12. Operational status by type of organizational unit of persons in mathematical employment, 1960---- Page 10 11 14 15 16 18 20 22 24 24 27 28 29 31 32 37 38 38 39 40 Table Table Table Table Table Table Table Table Table Table Table Table Table Table Table Table Table Table Table Table Table Table Table Table Table Table A–13. A–14. A–15. A–16. A-17. A–18. A–19. A—20. A–21. A–22. A—23. A–24. A—25. A–26. A–27. A–28. A-29. A—30. A–31. A–32. A—33. A–34. A—35. APPENDIX TABLES-Continued Mathematics education cited as minimum requirement for current position, by educational level and major subject field, for persons in mathematical employment, 1960- - --------------------- Major mathematical function of persons in mathematical employment, by employer, 1960--------- Median and quartile annual incomes of men and women in mathematical employment, by age and educational level, 1960------------------------------------------------------------------ Median and quartile annual incomes of persons in mathematical employment, by principal type of employer, educational level, and age, 1960------------------------------------------------- Median annual incomes of persons in mathematical employment, by age, educational level, and type of organizational unit, 1960--------------------------------------------------------- Distribution by income bracket and principal type of employer of persons in mathematical employ– ment, 1900---------------------------------------------------------------------------- Median annual incomes of supervisors and nonsupervisors in mathematical employment in private industry and in Government, by age and educational level, 1960----------------------------- Chief subject matter source of problems, by employer, of persons in mathematical employment, 1960– Major function, by chief subject matter source of problems, of persons in mathematical employment, Subject matter source of problems cited as first, second, and third most important by persons in mathematical employment, 1960--------------------------------------------------------- Mathematical fields used most by persons in mathematical employment, by employer, 1960- - - - - - - Mathematical fields used most, by chief subject matter source of problems, of persons in mathe- matical employment, 1960--------------------------------------------------------------- Mathematics courses cited by respondents as minimum requirements for their positions, by chief subject matter source of problems, 1960--------------------------------------------------- Mathematics courses cited by respondents as minimum requirements for their positions, by chief subject matter source of problems, for persons with a doctor's degree in mathematics, 1960------ Mathematics courses cited by respondents as minimum requirements for their positions, by chief subject matter source of problems, for persons with a master’s degree in mathematics, 1960–----- Mathematics courses cited by respondents as minimum requirements for their positions, by chief subject matter source of problems, for persons with a bachelor’s degree in mathematics, 1960- - - - Mathematics courses cited by supervisors as minimum requirements for a typical bachelor's degree level position under their supervision, by supervisor’s subject matter source of problems, 1960---- Mathematics courses cited by supervisors as minimum requirements for a typical bachelor's degree level position under their supervision, by mathematical field used most by Supervisors, 1960----- Minimum course requirements cited by persons in mathematical employment, by type of Organiza- tional unit in which they work, 1960------------------------------------------------------ Courses required for positions in mathematical employment for which formal training was hard to Secure, 1960--------------------------------------------------------------------------- Course deficiencies noted by supervisors among persons recently considered for typical bachelor’s level mathematics positions under their supervision, 1960------------------------------------ Deficiencies cited by supervisors in the mathematics training of post-World War II mathematics graduates, by educational level and field of employment, 1960-------------------------------- Deficiencies cited by supervisors in scientific or engineering training, other than mathematics, of post-World War II mathematics graduates, by educational level and field of employment, 1960–- . Percent distribution of respondents in private industry to survey of mathematical employment, 1960, and mathematicians, 1959–--------------------------------------------------------- . Percent distribution of respondents in private industry to survey of mathematical employment, 1960, and mathematicians, 1959, by size of employing company------------------------------ . Estimates of total employment in mathematical work other than teaching, 1960 Survey------------ 635761–62—2 Page 40 41 42 - 43 44 44 45 46 46 47 47 48 49 51 53 55 57 59 61 63 64 65 66 69 69 69 HIGHILIGHTS In response to a nationwide survey, about 10,000 persons who were engaged in professional mathe- matical work in industry, the Federal Government, and nonprofit organizations provided data on their personal and professional characteristics and the nature of their positions. These respondents represented approximately one-half of all the em- ployees concerned primarily with the application of mathematical techniques to problems of indus- try and Government in 1960. Chief employers of persons engaged in mathe- matical work were aircraft and electrical equip- ment manufacturers and the U.S. Department of Defense. Professional workers in mathematics tended to be concentrated in large companies. More than half the employees included in the survey were concerned with applying mathemati- cal techniques to research in the natural sciences and engineering; nearly a fifth were working on technical problems allied to production; and others were using mathematics in actuarial problems, economic research, business management, and in performing administrative functions. Nearly one-fourth of the respondents were working in computing laboratories, a type of organizational unit almost nonexistent 10 years earlier when few high-speed computers were in operation. About 15 percent were employed in engineering units, and nearly the same proportion were in mathematics units. In addition, 8 to 10 percent of all mathematical workers were attached to each of the following: General technical staffs, research laboratories, operations research units, and statistical units. Average (median) annual income from mathe- matical positions for persons at all educational levels was $8,500 in 1960. The average was $13,000 for those with the doctor's degree, $9,900 for holders of the master's degree, $7,700 for those with the bachelor's degree, and $7,900 for those who held no degree. Respondents in the insurance industry had the highest average income ($10,900); 15 percent had incomes of $20,000 or more. Employees in nonprofit organizations had second highest average earnings ($9,400). Even when employees of insurance companies were excluded, average incomes were higher in private industry than in the Federal Government— particularly for older, more experienced employees. Where comparisons were possible, average incomes of women in mathematical employment were well below those of men in the same age and education groups. One-third of all respondents with the doctorate earned professional income outside their major positions, and most of them earned the additional income by teaching in universities or colleges. - Approximately 94 percent of the survey respond- ents had at least a bachelor's degree, and one- third had earned an advanced degree—7 percent at the doctor's degree level and 26 percent at the master's degree level. The preponderance of employees whose highest degree was the bachelor's (61 percent) contrasts with the much higher educational attainment of the only large group of persons in mathematical employment not covered by this survey—those employed by universities or colleges. The rapid growth of mathematical employment in recent years is reflected in the age of the workers. Three-fifths of the respondents were under 35 years of age, a high proportion for an occupation made up almost entirely of college graduates. One respondent in seven was a woman and, as a group, women were younger than the men; 54 percent were under age 30. The youthfulness of the group, as a whole, was also reflected in the length of professional experience reported. Only 30 per- cent had more than 10 years of experience. A high level of demand in recent years for persons with mathematics skills has made it relatively easy for trained persons to enter mathematical employment and to change jobs. Job changing was more common for persons with fewer years of experience than for those who had been working for longer periods. An annual average of 20 job changes per 100 employees was reported by persons with fewer than 6 years of experience. However, about half the respondents reported only one employer in the last 10 years. Many employees apparently felt that their mathematical talents were being underutilized. Although employers were instructed to distribute questionnaires only to “professional personnel whose position . . . requires knowledge equal at least to that provided by a 4-year college course with a major in mathematics. ,” nearly one- fourth of the respondents said the performance of their duties required less training than the equiva- lent of a bachelor's degree. Holders of master's degrees were particularly dissatisfied with the level of mathematical work assigned to them; over half believed that the educational attainment required for their positions was lower than the level they had attained. When respondents were queried concerning the extent to which they used the level of mathematical education they considered as a requirement for their positions, about 4 out of 10 stated that they used it less than half the time. More than a third of all survey respondents had supervisory duties and, of this group, about 7 out of 10 supervised employees whose work was pri- marily mathematical. Of these supervisors the majority had responsibility for fewer than five employees. Median incomes of supervisors were about a third higher than those of nonsupervisors in private industry (excluding insurance) and in Government employment and approximately three-fourths higher in the insurance industry. CHARACTERISTICS OF PERSONNEL IN MATHEMATICAL EMPLOYMENT Employment in positions specifically concerned with the application of mathematical knowledge has been growing rapidly in the past decade. Although mathematics is one of the oldest of the scientific disciplines and has long been a necessary part of the background needed by physical Scientists and engineers, only in recent years have appreciable numbers of persons been hired in industry and Government primarily for their knowledge of mathematics rather than of the field to which it is applied. To learn more about these employees and the kinds of mathematical work they perform, ques- tionnaires were sent in early 1960 to a sample of selected companies in all types of private indus- tries and to all Federal Government agencies and nonprofit organizations known to employ mathe- maticians." Employers were requested to dis- tribute questionnaires to all professional personnel in full-time positions meeting the following criteria: (1) Performance requiring knowledge equal at least to that provided by a 4-year college course with a major in mathematics and (2) mathematics predominating over all other fields in either the nature of work done or in the educa- 1 For further information on the coverage of the survey, and insurance. tional prerequisites for the position. About 10,000 persons answered detailed questions con- cerning their age, sex, education, experience, income, and the characteristics of their work. These persons were engaged primarily in applying their knowledge of mathematics to research in the natural Sciences and engineering and to problems in the areas of production, administration, sales, A major group excluded from the survey was the mathematicians in educational institutions who were engaged in teaching and conducting research in mathematics. Employment in professional mathematical work other than teaching was estimated to be approxi- mately 20,000 in 1960. The detailed data throughout this report refer only to the replies from the 10,000 survey participants. However, these respondents are believed to represent to a reasonable extent the whole group as defined by the criteria cited above. (See appendix B.) Age and Sex - The relative youth of the survey respondents bears out other evidence that employment in mathematical work is growing rapidly. Three- fifths of the respondents were under 35 years of age—a high proportion for an occupation made up see appendix B. mostly of college graduates (table 1). Their TABLE 1.-Persons in mathematical employment, by age group and sea, 1960 - Total Men women Age group (years at nearest birthday) Number Percent Number Percent Number Percent All age groups------ ** * * = ** = = - * * * * * * * * * = <= 19, 718 100. 0. 8, 307 100. 0 1, 384 100. 0. Under 25 years------------------------------ 881 9. 1 525 6. 3 353 25. 5. 25–29-------------------------------------- 2, 529 26. 0 2, 122 25. 5 399 28. 8 30-34-------------------------------------- 2, 445 25. 2 2, 257 27. I 186 13. 4. 35-39-------------------------------------- 1, 684 17. 3 1, 518 18. 3 160 11. 6. 40-44-------------------------------------- 927 9. 5 810 9. 8 112 8. 1 45–49–------------------------------------- 578 5. 9 504 6. 1 73 5. 3. 50-54-------------------------------------- 386 4. 0 323 3. 9 61 4. 4. 55-59-------------------------------------- 181 1. 9 156 1. 9 25 1. 8 60–64-------------------------------------- 73 ... 8 64 ... 8 9 . 7 65 and over--------------------------------- 34 ... 3 28 ... 3 6 ... 4 Median age--------------------- — — — — — — — - - - - - 32.5 32.8 28.7 1 Excludes 264 persons who did not report age. Components may not add to totals because some individuals did not specify their sex. Chart 1. Women were concentrated in the younger age groups.... Percent distribution of men and women in mathematical employment, by age group, 1960 3O - § : 3 -: sº 20 |3 º º & - 3. 3. § : 10 F& &= º 3 º O º 3. : *— Sºğ &L Jºo º Under O 5 60 65 25 to fo to and 44 49 64 OVer AGE GROUP Source: Table 1. median age was 32.5 years, compared with a median age of 39 years in 1960 for all professional, technical, and kindred workers in the United States, and 38.8 years for engineers.” One in seven of the survey respondents was a woman. Women showed a greater concentration than did men in the younger age groups. One- fourth of the women—but only 6 percent of the men—were under 25 years of age; and 29 percent of the women, compared with 26 percent of the men, were in the 25- to 29-year age group (chart 1). One reason for the greater concentration of women in the younger age groups was the high proportion who began work in this recently fast- growing field without continuing their education beyond the bachelor’s degree. fewer than half as many women as men had ad- vanced degrees. Other reasons for the larger pro- portion of women in the young age groups are that many women marry and leave their jobs while still quite young and that very few women delay their education for military service. - 2 From unpublished data on the labor force compiled by the Bureau of the Census for the Bureau of Labor Statistics. Proportionately, Persons in mathematical employment in the manufacturing industries and in nonprofit organi- Zations were younger, on the average, than those in the nonmanufacturing industries, particularly insurance, or in the Federal Government (table 2). About two-thirds of those in manufacturing and in nonprofit organizations were under 35 years of age, compared with half of those in in- surance and Government. Since actuarial mathe- matics is fundamental to the operation of insurance plans, the insurance industry made use of mathe- matically trained persons early, with perhaps little change in the pattern of their employment in the last decade or two. The Federal Govern- ment has also employed mathematically trained persons for many years for its scientific and statistical activities. The large proportions of young people in mathematical employment in the manufacturing industries and in nonprofit or— ganizations reflect the rapid expansion in physical science research and military and civilian product development activities in the past decade and also the growing use of mathematically trained per- sonnel in solving production and other problems. 4 TABLE 2.-Age distribution of persons in mathematical employment, by broad industry group, 1960 Nonmanufacturing Age group (years at nearest birthday) Manufacturing Eederal Nonprofit Government | Organizations Insurance Other Number reporting "------------------------------------ 5, 522 919 545 2, 510 222 Percent distribution: - Total------------------------------------------- 100. 0 100. 0 100. 0 100. 0 100. 0 Under 25 years------------------------------------ 10. 4 7.7 8. 4 6. 9 9, 1 25–29--------------------------------------------- 29, 1 21, 9 22. 9 21. 6 24, 3 30-34--------------------------------------------- 27. 1 22.0 26. 8 21. 0 32.4 35-39--------------------------------------------- 18. 0 13. 4 17. 6 16. 8 21. 6 40-44--------------------------------------------- 7. 9 9. 9 10. 6 13. 0 8, 1 45-49--------------------------------------------- 4, 1 8. 3 6. 8 9. 2 2. 7 50-54--------------------------------------------- 2. 0 8. 5 2. 6 7. 1 1. 3 55-59--------------------------------------------- . 9 6. 3 2. 0 2. 4 ---------- - 60-64--------------------------------------------- 4 1. 5 1. 7 1. 2 ---------- 65 and over--------------------------------------- . 1 . 5 ... 6 ... 8 ... 5 Median age------------------------------------------ 31. 6 34. 1 33. 0 34, 7 32. 1 1 Excludes 264 respondents who did not report age. The growing use of electronic computers in the past 10 years, creating the need for mathematically trained persons to develop techniques for using the computers, has been a factor operating in many sectors of the economy to bring young persons into mathematical employment. The growing recognition of the contributions mathematical formulations can make to the solution of business management and military problems, as shown in operations research techniques, has also created a demand for more persons in mathematical work. Educational Attainment Ninety-four percent of the respondents who provided information on their education had college degrees and one-third of these had ad- vanced degrees—7 percent at the doctor's degree level and 26 percent at the master's degree level. Although 6 percent had not received a college degree, almost all of them (99 percent) had earned some college credits. A much smaller proportion of women than men employees had earned ad- vanced degrees. Only 2 percent of the women had doctorates and 15 percent had master's degrees, compared with 8 and 28 percent, respectively, for the men. (See chart 2 and appendix table A-1.) The proportion of survey respondents with advanced degrees is much smaller than if the survey had included college teachers of mathe- matics. It is perhaps also below that which would have been found for a group limited to persons with the title “mathematician.” ” However, it was the intent of this survey to exclude teachers and to include personnel in a broad range of mathe- matical positions regardless of occupational title, if the performance of their work required knowl- edge equivalent to a 4-year college course with a major in mathematics. Respondents with the master's degree were about 3% years older, on the average, and those with the doctorate were 6% years older, than respondents with only the bachelor’s degree. The median age for bachelor's degree holders was 30.6 years, compared with 34.3 for respondents with the master's degree and 37.0 for those with the doctorate (appendix table A-2). These age differ- ences can be accounted for only in part by the study time required to obtain graduate degrees, and many apparently gained some work experi- ence before attaining their advanced degrees. The relatively high average age (36.5 years) of respondents, who had not completed college indicates that they had more work experience, as a group, than the degree holders. It may also indicate decreasing opportunities for persons without college degrees to advance to professional mathematical positions. Major Subject Field of Degree About two-thirds of the survey respondents who had college degrees had majored in mathematics at their highest educational level; this includes 5 percent whose degrees were in statistics or actu- * Data from the National Register of Scientific and Technical Personnel, maintained by the National Science Foundation, indicated that 70 percent of the mathemati- cians registered in 1960 had advanced degrees. The Register contains a high proportion of college and univer- sity teachers, many of whom have the doctorate. For further information see Scientific Manpower Bulletin, No. 12, NSF 60–78, December 1960. 5 Chart 2. Proportionately, more men than women have advanced degrees.... Percent distribution of men and women in mathematical employment, by highest educational level attained, 1960 Doctor's degree Master's degree Bachelor's degree O 10 20 30 I | | 40 50 60 70 80 90 1OO i T I I ſ I . —l No degree Source: Table A-1. arial science (table 3 and appendix table A-3). In addition, almost one in four of those with advanced degrees in other fields had majored in mathematics at the bachelor's degree level, bringing the total with some degree in mathe- matics to 71 percent of all college graduates included in the survey. A shift from some other undergraduate major field had been made by one-fourth of about 2,000 respondents with advanced degrees in mathematics. (See appendix table A-4.) The response to this survey by a sizable proportion of persons who had obtained degrees in fields other than mathematics indicates that many persons in mathematical work in industry and Government must have knowledge of another subject matter area to which they apply their mathematics. A large proportion (70 percent) of those whose highest degrees were in subjects other than mathematics had specialized in engineering or the physical sciences—fields which have for many years been major areas for the application of mathematics. The remainder of the respon- dents with highest degrees in subjects other than mathematics were mainly from the fields of busi- ness and commerce, social science, and education. Although traditionally these fields, with the exception of education, have not been major users of persons with college-level mathematical skills, in recent decades they have developed an increas- ing need for these persons. For example operations research techniques are being used increasingly in business management; and research workers in the social sciences, life sciences, and psychology are applying probability theory to the design of experiments and are applying sampling theory to survey design. Extra College Credits Earned Survey participants were requested to indicate the number of semester hours completed after receipt of their highest degree or, if no degree was obtained, the total number of semester-hour credits earned. All but a few of the respondents who indicated they had no college degree had earned some credits (table 4). One-third of those without a degree who specified the number of credits earned had completed 3 or more years of college work, and 22 percent had completed 2 to 3 years. * However, the possibility that many of the “no degree” persons in mathematical employ- * Assuming 30 semester-hour credits as equivalent to a year of college work. 6 TABLE 3.−Major subject field of highest degree held by persons in mathematical employment, 1960 1 Major subject field Number Percent All subject fields-------------- 2 9, 249 100. 0 Mathematics----------------------- 6, 311 68. 2 Mathematics------------------- 5, 883 63. 6 Statistics---------------------- 351 3. 8 Actuarial Science--------------- 77 ... 8 Engineering------------------------ 1, 332 14, 4 Electrical and electronic--------- 385 4, 2 Mechanical-------------------- 344 3. 7 Chemical---------------------- 169 1. 8 Aeronautical------------------- 125 1. 4 Industrial--------------------- 77 8 Civil-------------------------- 73 ... 8 Other------------------------- 1.59 1. 7 Physics---------------------------- 484 5. 2 Chemistry------------------------- 142 1. 5 Other physical Sciences-------------- 101 1. 1 Business and commerce- - - - --------- 266 2. 9 Social sciences---------------------- 201 2, 2 Education------------------------- 119 1. 3 Humanities------------------------ 74 ... 8 Biological, medical, and agricultural sciences------------------------- 64 ... 7 Psychology------------------------ 59 ... 6 All other subjects------------------- 96 1. 1 1 Appendix table A-3 shows the distribution by major subject field of per- SOns at each educational level. * Excludes 733 respondents, of whom 566 had no college degree and 167 did not specify their educational attainment. ment will eventually attain a college degree appears unlikely, in view of the high proportion in the older age groups. Fifty-eight percent of these nondegree personnel were 35 years of age or older, 77 percent were at least 30 years old, and only 7 percent were under age 25 (appendix table A-2). Some respondents with college credits but no degree may have taken courses which had specific application to their work, making them very effective in a limited area of mathematical work. Half of the respondents with bachelor's degrees and 42 percent of those with the master's had earned some credits after receipt of their highest degree. Because there were many young persons in these groups, it is possible that a high propor- tion of them were working toward a higher degree. About one-seventh of those with the doctorate also had earned additional college credits; how- ever, only a few in this group had earned more than 30 extra credits. In addition to those reporting extra college Credits, 126 respondents—almost all with bache- lor's or master's degrees—volunteered the infor- mation that they had become “associates” or “fellows” of a professional actuarial society. Recognition as an associate or fellow is dependent on successful completion of examinations at college level and higher given by a professional actuarial Society. Since respondents were not specifically asked whether they had earned formal recognition as actuaries, the 126 who volunteered this infor- mation are undoubtedly only a small part of the total number of actuaries among the 926 persons in the insurance industry who responded to this survey." * The total number of professional actuaries in the United States is estimated to be approximately 1,700. See Occupational Outlook Handbook, 1961 edition, U.S. Department of Labor, Bureau of Labor Statistics. TABLE 4–Credits earned beyond highest specified educational level, for persons in mathematical employment, 1960 Highest educational level Number of Semester-hour credits earned All levels | Doctor’s Master’s | Bachelor’s | No degree Not; degree degree degree specified Number of persons reporting---------------------------- 1 9, 856 712 2, 496 5, 931 550 167 Percent distribution: • Total------------------------------------------ 100. 0 100. 0 100. 0 100, 0 100. 0 100. 0 No extra credits reported------------------------- 52. 8 85. 9 57. 8 50. 4 ... 7 96.4 Extra credits------------------------------------ 47. 2 14, 1 42, 2 49. 6 99. 3 3. 6 1-15--------------------------------------- 15. 6 3. 9 12. 6 19, 5 7, 3 |-------- 16-30-------------------------------------- 10. 5 2, 4, 10. 0 11. 9 10. 7 -------- 31-00-------------------------------------- 6. 3 ... 7 6, 9 5. 9 16.9 |-------- 61-90-------------------------------------- 2. 3 3 2. 1 1. 2 17. 6 6 91 or more---------------------------------- 2. 1 ... 3 ... 8 ... 7 26.4 6 Credits earned but number not specified-------- 10.4 6. 5 9. 8 10. 4 20.4 2. 4 1 Excludes 126 persons who reported they had passed actuarial examinations leading to fellowship or associateship in a professional actuarial society, but did not report number of credits earned. 635761—62—3 Chart 3. Advanced degrees are earned over a broad age range.... Percent Age at receipt of highest degree by persons in mathematical employment, 1960 40 Bachelor's / degree 30 2 *s * \ | | | | | | \ 2O |– | ! | | Mdster's | ! d | \ / egree V - - - - - - ſ ... “.... * \ • 10 ! #-A-H2 **. ! : N-X | Doctor's ! : Y ~ degree I & N. I - | / .” O /1.** A -- f I Irs---ºº:::::::::::::FE::firrºr- T W I I y I º y | I l 15 2O 25 3O 35 40 45 50 55 Age (to nearest birthday) Source: Table A-5. Age at Receipt of Degree Among the respondents who were college gradu- ates, the median age for receipt of their highest degree was 22.6 years for the bachelor’s degree, 26.2 for the master’s degree, and 28.7 for the doctorate. It should be noted that the median age for receipt of the master’s degree reported here refers only to those whose highest degree was the master's. It is quite likely that respondents with doctorates had earned their master's degrees at an earlier age, on the average, than had those whose highest degree was the master's. About 58 percent of all respondents with college degrees earned the bachelor's degree at ages 21 to 23, and only 6 percent earned that degree after age 29. There was a much wider span in the ages at which advanced degrees were earned. Fifty- seven percent of those whose highest degree was the master's had earned the degree at ages 23 to 27, and 67 percent of the doctorates were earned in the ages 25 to 31 (chart 3 and appendix table A–5). Respondents who had attained advanced degrees had, on the average, earned their undergraduate degree at a somewhat earlier age than the average for all college graduates in the survey. Median age at receipt of the bachelor's degree, for those whose highest degree was the doctorate, was 22 years; for those at the master's degree and bache- lor's degree levels the medians were 22.3 and 22.6 years, respectively. Although these variations in median age for receipt of the bachelor's degree were slight, there were large differences in the proportions of respondents at each level of educa- tional attainment who had received their bache- lor’s degree before they were 22 years of age—38 percent of those with the doctorate, 25 percent of those with the master's, and 18 percent of those whose highest educational level was the bachelor's degree. (Detailed data are shown in appendix table A-6.) Timelapse Between Degrees The majority of the survey respondents with advanced degrees apparently continued their edu- cation beyond the bachelor’s degree on a part-time basis only or interrupted their education for work or other reasons. Almost 10 percent of the re- spondents with the doctorate had earned it in the 3 years immediately following receipt of the TABLE 5.—Distribution of persons in mathematical employ- ment, by number of years between baccalaureate and advanced degrees, 1960 All respondents Mathematics ImajorS Years between degrees Bache- | Bache- | Bache- | Bache- lor’s to lor’s to lor’s to lor’s to doctor's master’s doctor’s master’s degree degree | degree degree Number reporting *-------- 683 |2, 422 || 426 1, 527 Percent distribution: Total.-------------- 100. 0 || 100. 0 || 100. 0 || 100. 0 Under 2 years--------- 1. 3 || 23.4 | 1. 2 | 24, 1 2-------------------- 1. 5 || 23. 1 2. 1 23. 8 3-------------------- 6. / | 11, 9 5, 6 11. 7 4-------------------- 16. 6 || 10. 8 || 15. 8 10. 5 5-------------------- 15. 9 7. 3 || 15. 8 7. 1 6-------------------- 10. 0 5. 8 9, 2 5. 4 7-------------------- 9. 7 5. 4 9, 2 5. 3 8-------------------- 8. 8 3. 1 8. 4 3. 1 9-------------------- 7. 8 2. 0 8. 4 1. 8 10------------------- 5. 3 1. 7 5. 6 1. 8 11------------------- 2. 9 1. 0 2. 8 ... 7 12------------------- 2. 8 . 9 3. 9 . 9 13------------------- 1. 9 . 7 || 2. 1 ... 7 14------------------- 1. 6 . 4 1. 4 . 5 15.------------------- 1. 5 . 5 2. 1 . 4 16------------------- 1. 6 ... 4 1. 6 ... 3 17------------------- . 9 ... 2 ... 7 ... 3 18------------------- ... 6 ... 2 9 ... 3 19------------------- ... 6 ... 2 9 . 3 20 and over----------- 2. 0 1. 0 2. 3 1. 0 Average number of years - between degrees— — — — — — — — — 7. 3 4. 0 7. 6 4. 0 1 126 respondents did not provide information on this topic. baccalaureate, and fewer than one-fourth of the respondents with the master's degree received it in the year following receipt of the bachelor's degree. The mean timelapse between receipt of the bachelor's and doctor's degrees for all who had received the doctorate was 7.3 years. When con- sideration was limited to those with a mathematics major at the doctorate level, the timelapse between the two degrees averaged somewhat longer—7.6 years.” In addition to the 10 percent of all re- spondents who had earned the doctorate within 3 years after receipt of the bachelor's degree, about one-third reported a lapse of 4 or 5 years between these degrees, and 22 percent reported 10 or more years (table 5). • The National Academy of Sciences-National Research Council found that the mean timelapse between bacca- laureate and doctor's degree for mathematics majors who had earned their doctorate in calendar years 1958 and 1959 was 8.1 years. The study also reported that the 1958–59 group had an average (median) of 3.8 years of predoctoral professional experience. See The Science Doctorates of 1958 and 1959, National Science Foundation, NSF 60–60, 1960, p. 15. The mean timelapse between receipt of the bachelor's and master's degrees was 4 years for respondents whose highest degree was the master's. Approximately the same proportions (23 percent) earned their advanced degree in the first and second years following receipt of their under- graduate degree; 7 percent reported 10 years or more between receipt of the bachelor's and master’s degrees. Professional Experience Recent rapid growth of mathematical employ- ment in private industry and in Government is indicated by the high proportion of employees with relatively few years of professional experi- ence. Forty-one percent of the survey respond- ents reported less than 6 years of professional experience, 29 percent had 6 to 10 years, and only 11 percent reported 20 or more years of experience. (See chart 4 and appendix table A-7.) Since the respondents were asked to report all of their professional experience, the length of time spent Chart 4. Four out of ten employees had less than 6 years of professional experience.... -Employees 1,000 800 600 400 200 O § % º 38& 33 º B& ºil.’ Years of ſº - - - TV. ' - . §ºncé 5 10 15 20 25 30 35 | as of 1960. } +. 1960 1955 1950 1945 1940 1935 1930, 1925 - Assumed year of first employment *Less than 6 months Source: Table A-7. TABLE 6.--Distribution of persons in 1960 field of mathe- matical employment by field of prior experience Field of 1960 employment Field of prior experience r Private Federal Nonprofit industry Govern- Organiza- ment tion Number reporting---------- 7, 171 2, 586 225 Percent distribution: Total--------------- 100. 0 100. 0 100. 0 Current field only-------- 63. 0 59. 8 24. 7 Other fields 4------------ 37. 0 40. 2 75.3 Private industry-------|-------- 26. 9 54. 3 Government----------- 22. 4 |-------- 26. 9 University or college--- 15. 6 13. 2 35. 0 Nonprofit organization-- 4, 7 6. 3 |-------- Self-employment------- 2. 6 4. 2 3. 6 1 Components will add to more than totals because some respondents had experience in more than one field. in mathematical employment may be somewhat less than total years reported. About 40 percent of the respondents reported professional experience in a field of employment other than the one in which they were engaged at the time of the survey. Twenty-two percent of those working in private industry had pre- viously worked for the Government, and 16 percent had previously been employed by colleges or universities (table 6). Of the Government employees, 27 percent had worked in industry and 13 percent had been employed by univer- sities or colleges. Small proportions of those working in industry or Government at the time of the survey had previously been self-employed or had worked for nonprofit organizations. Three-fourths of those working for nonprofit organizations in 1960 reported some other pro- fessional experience, chiefly in industry. The distribution by years of experience is of interest when shown with a chronological scale of “assumed year of first employment.” (See chart 4.) The assumed year of first employment was derived by subtracting the reported years of experience from 1960—the date the survey was conducted. For example, those with 1 year of experience were assumed to have started work in 1959; those with 2 years in 1958, etc. This procedure obviously does not measure accurately the year of entry into mathematical employment of all respondents. About 16 percent of the re- spondents reported experience in teaching (a field excluded from this survey) before taking their current position in industry, Government, or a nonprofit organization. Moreover, some had undoubtedly interrupted their working experience for further education, military service, or for other reasons. Although the influence of these factors is not known, the pattern of the distribution suggests that the recession of 1948–49, 1953–54, and 1957–58 slowed, but did not halt, the growth of employment of persons in mathematical work. The distribution also suggests that employment growth began its spurt in 1948 with the increased supply of post-World War II graduates and before electronic computers became an important factor. Professional Work Outside Major Position In addition to their major position, about one in seven of the respondents had other income- producing professional work. Respondents with advanced degrees were more likely to have secondary professional employment than were those at lower educational levels, and college or university teaching was the most common pro- fessional activity outside their major position (table 7). About 5 percent of the “no degree” respondents earned outside income from consulta- tion. (See also section on income, page 27.) TABLE 7.-Percent of persons in mathematical employment who had professional work outside their major position, by educational level and source of secondary employment, 1960 Highest educational level Source of Secondary employment - All levels Doctor’s Master’s Bachelor’s No degree degree degree degree Percent reporting secondary employment 1 - - - - - - - - - - 14. 7 33. 4 21. 0 9. 9 14. 0 College or university teaching--------------------------- 5. 4 20. 2 10. 1 2. 1 . 9 High school teaching----------------------------------- 4 ---------- ... 4 . 4 ... 4 Other teaching----------------------------------------- 2. 3 2. 4 2. 7 2, 2 1. 7 Writing----------------------------------------------- 7 2. 2 . 9 ... 4 . 9 Consultation------------------------------------------ 2. 6 3. 7 2. 8 2. 1 5. 4 Other sources and combinations of the above sources------- 3. 3 4. 9 4. 1 2. 7 4. 7 1 Includes income-producing professional employment only. TO TABLE 8.—Annual rate of job changes by persons employed in mathematical work, by educational level and experience, 1 950–60' Highest educational level Years of experience All levels Doctor’s Master’s Bachelor’s No degree degree degree degree Under 3 years----------------------------------------- 21 20 34 19 13 3-5--------------------------------------------------- 19 29 22 17 16 6-10-------------------------------------------------- 15 20 18 14 14 11-16------------------------------------------------- 12 18 13 10 13 16-20------------------------------------------------- 8 12 10 7 6 21-25------------------------------------------------- 7 10 8 6 6 26-30------------------------------------------------- 4 10 4 4 ---------- 31-36------------------------------------------------- 4 ---------- 7 3 ---------- 1 Annual average of changes made in last 10 years or in a shorter period if respondent had less than 10 years of experience. Job Changing, 1950–60 A high level of demand in recent years for mathematicians has made it relatively easy for well-trained individuals to change employment as well as to make the initial entry into the pro- fession. The Bureau of Labor Statistics, for example, reported in 1957, 1959, and 1961 that the employment outlook for mathematicians was very favorable, particularly for holders of doc- tor's degrees." In January 1961, Professor Samuel S. Wilks, Chairman of the Conference Board of 7 U.S. Department of Labor, Bureau of Labor Statistics, Occupational Outlook Handbook, 1957 edition, p. 135, 1959 edition, p. 132, and 1961 edition, p. 150. Chart 5. The majority have worked for no more than one employer.... Distribution of persons in mathematical employment, by years of experience and number of employers in last 10 years, 1960 l Five or more Percent 100 o o o O 9 o © 60 40 © e e e & a º e e e º e e ‘s e e º, e º e e 8O © tº e e e -e ºs e tº e º e tº e º 'º tº © & © tº tº s “º º tº gº tº º º tº, tº e º e e º º * , " . •". ge º e O Under 3 6 11 16 21 26 31 36 to fo to to to to to to 3 5 1O 15. 20 25 30 35 40 t © Years of experience | Covers a shorter period for respondents with fewer than 10 years of experience. employers z -I -ms js four employers Three employers o Two employers © e º e tº º º tº º e º 'º e e º e e e - - to e º 'º tº 6 - - & º E tº One employer. ~- Source: Table A-8. 11 the Mathematical Societies, stated, “The demand for mathematicians is now tremendous . . .” “ Employers replying to the 1960 survey reported that about 1 in 10 of their mathematical positions was unfilled. r Respondents to this survey who had less than 6 years of experience reported an average of about 20 job changes per 100 employees per year over the period of their experience (table 8).” Persons with long experience reported relatively few employment changes during the 10-year period 1950–60. The greatest amount of job- changing activity at all experience levels took place among those with advanced degrees. For example, holders of doctorates with 3 to 5 years of experience reported 29 job changes per 100 em- ployees per year, and those with 21 to 30 years of experience had maintained an average of 10 job changes per 100 employees each year for the last 10 years. On the other hand, job changes for bachelor's degree holders at the same experience levels were 17 and 5, respectively. Apparently employees with a high level of educational attain- ment had more opportunities to improve their employment status by shifting from one employer to another. More than half the employees in the survey reported only one full-time professional position in the last 10 years. This includes respondents with less than 10 years of experience. About one- fourth had changed employers once and the remain- ing fourth had changed at least twice (chart 5 and appendix table A-8). Small proportions in each experience group had worked for five or more employers, reaching 7 and 8 percent, respectively, among those with 6 to 10 and 11 to 15 years of experience. * The Washington Post, Washington, D.C., January 26, 1961. - ° The data in table 8 understate the amount of job changing. A number of respondents apparently mis- understood the survey question on this topic and omitted their current employment in counting the number of positions they had held in the last 10 years. T2 CHARACTERISTICS OF MATHEMATICAL POSITIONS, 1960 Information on conditions of employment and the kinds of work performed in mathematical positions was obtained by asking survey partici- pants to indicate the type of organizational unit in which they were working; whether they were primarily administrators, supervisors, or mathe- matical practitioners; what educational level they believed necessary to perform their work; the income from their position; and the mathematical Content of their work. Information was also obtained on the industry or agency in which employees worked and, if respondents worked in private industry, on size of company. This sec- tion of the report discusses their employment environment and relates educational attainment and age of respondents to the various aspects of employment. Type of Employer The survey of mathematical employment in private industry, Federal Government, and private nonprofit organizations covered all major em- ployers except educational institutions." The chief employers of persons in mathematical work in 1960 were aircraft and parts and electrical equipment manufacturers and the U.S. Depart- ment of Defense. Other manufacturing industries employing substantial numbers primarily for mathematical work included machinery (except electrical), petroleum products (including extrac- tion), and chemicals and allied products. Among nonmanufacturing industries, insurance was the largest employer, but many employees in mathe- matical work were also found in engineering and architectural services, telecommunications, and utilities.” * State and local governments were also omitted. Only about 350 mathematicians were employed in State govern- ment agencies in 1959. (See Employment of Scientific and Technical Personnel in State Government Agencies, Report on a 1959 Survey, National Science Foundation, NSF 61–17, 1961.) It is believed that employment of persons for mathematical work in local governmental units is very small. Educational Level by Type of Employer. The proportion of respondents with advanced degrees was much higher in nonprofit organizations (56 percent) than in private industry (34 percent) or in the Federal Government (28 percent). (See table 9.) However, substantial variation from the overall average in the proportion of employees at each educational level appeared for some industries and agencies. For example, only about 1 percent of the personnel in mathematical em- ployment in the insurance industry had earned doctorates, compared with 17 percent in the chemical industry. Similarly, about 3 to 4 percent each of the Department of the Army and National Aeronautics and Space Administration employees in mathematical work held the doctor's degree, compared with 20 percent of these employees in the U.S. Department of Commerce. Data col- lected in this survey do not provide a basis for a systematic explanation of the varying proportions at each educational level in different industries and Government agencies. Nevertheless, dif- ferences in the proportion of employees at each educational level are likely to be related to the type of mathematical work performed. In the insurance industry, where more than three-fourths of the professional workers in mathematical employment were concerned with actuarial mathematics, only about one in five had received an advanced degree. The primary qualification in this field is achieved not by ob- taining advanced degrees but by successfully completing the associateship and fellowship ex- aminations given by professional actuarial Societies. On the other hand, in the chemical industry, where companies carry on a great deal of research which requires personnel with a high level of * Although the coverage of this survey was not limited to persons with the professional title of mathematician, the distribution of respondents by industry was approxi- mately the same as that for mathematicians found in a survey of scientific and technical personnel in industry in 1959. Data from the two surveys are compared in appendix B, tables B–1 and B-2. TABLE 9–Educational level of persons in mathematical employment, by employer, 1960 Percent distribution by educational level Number Employer reporting Total Doctor's Master's Bachelor’s | No degree degree degree degree All employers------------------------------------- 1 9, 815 100 7. 2 25. 7 61. 3 5. 8 Private industry----------------------------------------- 7,098 100 7. 4 26. 5 61. 4 4. 7 Aircraft and parts------------------------------------ 1, 961 100 5. 9 25. 1 63. 3 5. 7 Transportation equipment (except aircraft) - - - - - - - - - - - - - 208 100 6. 7 28. 4 62. 5 2. 4 Electrical equipment--------------------------------- 1, 226 100 8. 7 27. 5 58. 9 4. 9 Machinery (except electrical)-------------------------- 601 100 11. 8 26, 9 55. 5 5. 8 Professional and Scientific instruments.------------------ 186 100 10. 8 29. 6 58. 0 1. 6 Other durable manufacturing-------------------------- 521 100 4. 8 31. 0 58. 6 5. 6 Petroleum products and extraction--------------------- 451 100 12.4 27, 1 58. 9 1. 6 Chemicals and allied products------------------------- 317 100 17. 1 32. 6 47. 1 3. 2 Other nondurable manufacturing----------------------- 169 100 10. 7 31.4 53. 8 4, 1 Insurance------------------------------------------- 909 100 ... 8 20. 7 73. 3 5. 2 Other nonmanufacturing------------------------------ 549 100 6.4 27. 3 63. 0 3. 3 Federal Government------------------------------------- 2,493 100 5. 8 22. 5 62. 4 9. 3 Army---------------------------------------------- 929 100 3. 1 19. 4 60. 2 17. 3 Navy----------------------------------------------- 577 100 7. 1 22. 5 66. 9 3. 5 Air Force------------------------------------------- 453 100 5. 3 26. 5 61. 1 7. 1 National Aeronautics and Space Administration--------- 231 100 3. 5 16. 0 76. 2 4. 3 Commerce------------------------------------------ 104 100 20, 2 21. 1 57. 7 1. 0 All other agencies------------------------------------ 199 100 10. 0 35. 7 50, 8 3. 5 Nonprofit organizations----------------------------------- 224 100 20. 5 35. 3 43. 3 . 9 1 Excludes 167 respondents Who did not specify educational level. educational attainment in the sciences and mathe- matics, half the employees had advanced degrees. The high proportion (76 percent) of National Aeronautics and Space Administration employees whose highest degree was the bachelor's is related to the agency’s need for personnel to work on large-scale scientific data reduction problems in connection with the space program. Employees with a mathematics major at the bachelor’s level are usually hired to program satellite track- ing and performance characteristics for computers and to operate the computers. The relatively low proportion (23 percent) with advanced degrees in the Department of the Army is due, in part, to the fact that questionnaires were distributed to persons classified as management analysts, as well as to mathematicians, mathe- matical statisticians, and actuaries.” In the U.S. Department of Commerce, the relatively high proportion (20 percent) of employ- ees with the doctor's degree is accounted for largely by respondents from the Bureau of Standards and its Applied Mathematics Division. At the time this report was written, 16 of the 48 * Only reports from management analysts whose posi- tions appeared to meet the minimum criteria for inclusion in the survey were used in the tabulations. Nevertheless, it was evident that most management analyst jobs did not require extensive mathematical training. mathematicians at the Bureau of Standards held the doctorate. The high proportion (56 percent) of advanced degrees among persons in mathematical work in nonprofit organizations is related to the fact that most of such organizations in the survey were re- search centers or institutions which are heavily engaged in physical science research. Size of Employing Company The majority of persons hired primarily for mathematical work in private industry are em- ployed by large companies. About 80 percent of the respondents to the survey worked in companies with a total employment of more than 5,000. However, the employee replies tended to overrep- resent such employment. Other surveys covering mathematicians indicate that the proportion in companies of 5,000 or more employees is closer to 73 percent. (See appendix B.) Mathematicians are concentrated to a greater extent in large com- panies than are all engineers and scientists as a group. Only about 53 percent of all scientists and engineers in the Nation's industry in January 1959 were in companies with 5,000 or more employees.” 4 Scientific and Technical Personnel in American Indus- try, Report on a 1959 Survey, National Science Foundation, NSF 60–62, p. 31. T 4. TABLE 10.-Educational attainment of persons in mathematical employment in private industry, by size of company, 1960 Percent distribution by educational level Number Company size (number of employees) reporting Total Doctor’s Master’s | Bachelor’s | No degree degree degree degree All size groups------------------------------------ 17, 098 100 7. 4 26. 5 61. 4 4. 7 Under 1,000 employees----------------------------------- 264 100 3. 8 20. 8 67. 8 7. 6 1,000 to 4,999 employees - - - - - - --------------------------- 1, 130 100 5. 3 26. 1 63. 6 5. 0 5,000 to 24,999 employees – º – - - - - - - - - - - ------------------- 1,989 100 7. 4 28. 0 59. 8 4. 8 25,000 or more employees--------------------------------- 3, 715 100 8, 2 26, 2 61. 3 4. 3 1 Excludes 73 respondents who did not specify educational level. Educational Level by Size of Company. The pro- portion of persons with advanced degrees was greater, on the average, in large than in Small firms (table 10). Only about 4 percent of the persons in mathematical employment in companies with total employment under 1,000 had received the doctorate, whereas 8 percent of those in firms with 25,000 or more employees had obtained that degree. Conversely, three-fourths of the respond- ents in the smallest companies, but only two-thirds in the largest companies, had less than the master's degree. Organizational Unit of Employment The organizational units in which survey re- spondents were working in 1960 reflected both the old and the new environment of mathematical employment. More than 40 percent of the re- spondents were in computing laboratories, mathe- matics units, and operations research units—types of organizational units which, except for a few pioneering groups, did not exist prior to World War II. In research laboratories, engineering units, and statistical units, mathematical work has long had an important role. Even in these long- established types of organizations, however, the employment pattern for mathematical work has been changing. Distinctive age and education patterns appeared for employees in a few types of organizational units. A high proportion of young persons were in computing laboratories; three-fourths were under 35 years of age in 1960. On the other hand, in administrative units more than three-fourths of the mathematical workers were over 35 years, and persons working on general technical staffs and in statistical units also were older, on the average, than all respondents to the survey. (See chart 6 and appendix table A–9.) Persons with advanced degrees, particularly at the doctorate level, were employed in higher proportions in research labo- 635761–62—4 ratories, operations research units, and mathe- matics units. Relatively few employees in pro- duction, administrative, and engineering units and in computing laboratories had received the doc- torate. (See chart 7 and appendix table A-10.) Computing Laboratories. One of the great stimu- lants to the recent growth of mathematical em- ployment has been the rapid increase in the use of high-speed electronic data computers in indus- try and Government. Almost one-fourth of the survey respondents were working in computing laboratories (table 11), and much additional math- ematical employment has been generated in other types of organizational units because computers facilitate the use of mathematical ability at all levels of difficulty. Chart 6. Computing laboratories had the youngest group of mathematical workers; administration units the oldest.... 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'... *.*.*.*.*.*.*.*.*.*.*.*.*.*.*.*.*.*.*.*.*.*.*.*.*.*.*.*.*.*.*.*.*.*.*.*.*.*.*.*.*.*.*.*.*.*.*. 2 / / / / / sº º *—a—º-–a–A––a–a–4a–A–a–a–s—a-a-—A-a-—n—a-—a--a – a – º –in–in–in–in–a–s. –º ––a -- in a R fº ſh–A–F– –F– º –F–8—?–a–sº *—ſº-ſº ſº -in A Z Z Z.Z. A & Source: Table A-10. The first high-speed electronic computer, the ENIAC, built by the University of Pennsylvania for the Army’s Ballistics Research Laboratory, was completed in 1946. Two more electronic computers were built by the Bureau of Standards and the International Business Machines Corpora- tion before the first commercially available machine (the Univac) was installed for the Bureau of the Census in 1951. By July 1960, almost 11,000 computer installations were in use.” Electronic computers, because of their speed of operation, make possible the solution of many mathematical statements of engineering, scientific, and business problems which could not be solved economically by previous methods. Persons with mathematical training are employed to state the problems properly, and others translate the in- * Testimony by Dr. A. V. Astin, Director, National Bureau of Standards, pp. 572–589, and Robert W. Burgess, Director, Bureau of the Census, p. 72, in “Automation and Technological Change,” Hearing before the Subcom- mittee on Economic Stabilization of the Joint Committee on the Economic Report, (84th Cong.), 1955, and by John Diebold in “New Views on Automation” before the Sub- committee on Automation and Energy Resources of the Joint Economic Committee, (86th Cong.), 1960, p. 115. formation into machine language for processing (programing) and may operate the equipment. Highly trained mathematicians also assist in designing the many models of computers being developed. Although the availability of high- speed computers facilitates the utilization of high- level mathematical ability of employees, the majority of persons working directly in computer units are young graduates with bachelor's degrees only (charts 6 and 7 and appendix tables A–9 TABLE 11.-Mathematical employment, by type of organizational unit, 1960 Organizational unit Number Percent, All units--------------------- 19, 929 100. 0 Computing laboratory--------------- 2, 288 23. 0 Engineering unit------------------- 1, 514 15. 2 Mathematics unit ------------------ 1, 229 12. 4 General technical Staff- - - - ---------- 956 9. 6 Research laboratory---------------- 875 8. 8 Operations research unit------------- 791 8. 0 Statistical unit--------------------- 781 7. 9 Administration unit----------------- 358 3. 6 Production unit-------------------- 167 1. 7 Other----------------------------- 970 9. 8 1 Excludes 53 persons who did not specify organizational unit. T6 and A-10). Training at the bachelor's degree level is adequate in most cases for programing and machine operation. Not all computing units make use of math- ematically trained persons, particularly if the work done on the machines is limited to the processing of business data such as payrolls and billings. On the other hand, many of the re- spondents who identified themselves with other types of organizational units had computer time available to them for problem solving or data processing. Mathematics Units. The first, formally estab- lished, applied mathematics unit in industry or Government was the Bell Telephone Laboratories group which was organized about 30 years ago. In 1938, the WPA-financed Mathematics Tables Project was operating in New York. It sub- sequently became part of the Applied Mathematics Division of the National Bureau of Standards, which was organized in Washington, D.C., in July 1947. - Over the years, several groups now a part of the U.S. Department of Defense have established mathematics units. Important precursors of the separately identified mathematics units were groups concerned with ballistics research and computation for both the Navy and Army, The Exterior Ballistics and Computation Laboratory at the Naval Proving Ground, Dahlgren, Va., was established in the early years of World War II. One of the first major automatic computer ap- plications took place there using an electro- mechanical computer completed by Harvard University in 1943. As noted earlier, the first electronic computer, the ENIAC, was built for the Army’s Ballistics Research Laboratory, which also was organized in the early 1940's. Federal Government groups specifically iden- tified as mathematics units include the Air Force's Applied Mathematics Research Branch at Wright Aeronautical Development Center, established in 1948; the Mathematical Sciences Division of the Office of Naval Research, organized in 1945; the Mathematics Department of the Naval Ordnance Laboratory (about 1949); the Applied Math- ematics Laboratory at Navy's David Taylor Model Basin (1952); and Army's Mathematical Sciences Division in the Army Research Office. The Army also supports other mathematics groups which are under the management of universities. - of age. matics units with the doctorate was 13 percent, Although information on the establishment of separately identifiable mathematics units in pri- vate industry is obscure, such units probably were not being established widely in industry until the mid-1950's. Respondents from 135 different companies reported that they worked in mathe- matics units in 1960. As in computing labora- tories, a high proportion of the employees in these units were young—two-thirds were under 35 years The proportion of employees in mathe- much higher than the 3 percent found in comput- ing laboratories and second only to the proportion of employees with doctorates in research laboratories. Operations Research Units. Operations research may be broadly defined as a logical and thorough approach to defining and seeking solutions to problems—of any character—using knowledge and theory from all the subject fields that bear on the problem. Mathematical methods are central to much operations research, but advanced mathe- matical techniques are not always necessary, and specialists in a wide variety of subject matter fields work on operations research teams. The first operations research unit in the United States was established by the Navy in 1942, under Massachusetts Institute of Technology manage- ment. It followed successful use of operations research techniques in the British war effort and was soon duplicated by many other operations research groups in the Departments of the Navy and Air Force and, to a lesser extent, in the De- partment of the Army. In the U.S. Department of Defense, operations research groups work on strategy, tactics, logistics, and other types of problems. The operations research approach to problem solving has also been adopted widely by private business. Respondents from 144 com- panies and 7 nonprofit research organizations reported they were working in operations research units." - Respondents working in operations research units reported the smallest proportion with no degree (2 percent), the highest proportion with the master's degree (35 percent), and a relatively high proportion of doctorates (12 percent). Of the respondents working in operations research * For a discussion of the early history of operations research, see Operations Research for Management, edited by J. F. McCloskey and F. N. Trefethen, Baltimore, Md., Johns Hopkins Press, 1954. - 17 units, 42 percent had a major subject other than mathematics at their highest educational level. Research and Engineering Units. Mathematics is, of course, a fundamental part of the physical sciences and engineering. In the past, almost all necessary mathematical work in engineering and physical science research units was performed by employees who were primarily engineers, physi- cists, and chemists. In some organizations, only one or a few mathematicians were employed as consultants. In recent years, however, the re- search problems have become so complex and the mathematical models of them so difficult that engineering and research groups have found it necessary to hire increasing numbers of persons primarily trained in mathematics. About one-fourth of the respondents to this survey were working in these traditional areas of applied mathematics—15 percent in engineering units and 9 percent in research laboratories or units. Of those in research laboratories 18 per- cent had received the doctorate, the highest pro- portion in any of the nine specified types of organ- izational units. On the other hand, fewer than 4 percent of the respondents in engineering units held the doctorate. Statistical Units. Statistics, as a science based on the mathematical theory of probability, afforded employment to pioneers in statistical theory in the calculation of gambling odds. In recent years there has been a growth in the applica- tion of mathematical statistics to physical and biological phenomena and to business manage- ment problems by the use of statistical theory in the design of experiments, in quality control, and in evaluating the likely result of proposed plans of action. Much of the work in statistics is, however, in the social sciences, and such studies are more common in Government than in private industry. About 12 percent of the Government employee respondents to the 1960 survey were in statistical units, but only 6 percent of the respondents in private industry were in such units. The use of statistics is not limited to employees in statistical units, and many persons who identi- fied themselves with other organizational units undoubtedly made use of statistical theory in their work. Other Units. Other organizational units sepa- rately tabulated in the survey, and which employ persons for mathematical work, include general technical staffs, administration units, and pro- .-” duction units. Less than 2 percent of the re- spondents were employed in production units and fewer than 4 percent were in units specifically concerned with administration. Few persons doing mathematical work in these units had received the doctorate. Persons in administrative units were markedly older, on the average, than employees working in other groups. It is likely that many persons who had difficulty in identifying themselves with other specific organizational units classified themselves in the general technical staff unit. One-third of the respondents checking this category were in the insurance industry. Many of these were actuaries who may have found the specific organizational units listed in the questionnaires not descriptive of their situation. Operational Status Slightly more than one-fourth of all survey respondents were primarily administrators or supervisors, and 66 percent were mathematical practitioners. The majority of the practitioners, particularly those who had majored in mathe- matics, were working in groups either with other mathematicians or, to a lesser extent, as members of teams made up of nonmathematicians (table 12). TABLE 12.-Operational status of persons in mathematical employment, by major field of education, 1960 Major field of education Operational status All fields | Mathe- Other matics Number reporting-------------- 19, 721 | 6, 530 3, 191 Percent distribution: Total------------------- 100. 0 || 100. 0 || 100. 0 Administrator------------- 7. 3 6. 2 9. W Supervisor---------------- 18. 5 15. 8 || 24, 2 Mathematical practitioner--- 65. 5 || 70.3 || 55.4 Working primarily with other mathematicians– 23. 8 28. 7 13. 8 Working individually with little or no con- Sultation with others--| 13.0 13. 7 11. 4 Working as an individ- ual consultant to others-------------- 10. 6 10. 0 11. 7 Working as a member of a team made up mostly of nonmathe- maticians----------- 18. 1 17. 9 18. 5 Other--------------------- 8. 7 7.7 10. 7 1 Excludes 261 respondents who did not specify operational status or educa- tion major. T 8 The insurance industry had an unusually high proportion of mathematical workers in administra- tive and supervisory positions—28 and 30 percent, respectively. Furthermore, almost 1 in 10 of all insurance industry respondents was either a vice president or assistant vice president of his firm. As reported earlier, most insurance employees in the survey indicated they were concerned with actuarial work. Another unique situation with respect to administrative personnel was that reported in the Government sector by employees of the Department of the Army; one in eight of these respondents reported that he was an ad- ministrator. Almost all in this operational status had the job title “management analyst,” an employee classification which the Department of the Army, but no other Government agency, considered within the scope of the survey. Management analysts in the Department of the Army were concerned mainly with evaluating progress and making special studies of major programs of the Department. In order to examine the characteristics of workers in different types of positions, operational status was related to age, educational level, major subject, and type of organizational unit in which respondents were employed. The factor most closely identified with operational status appeared to be the age of respondents. Age. The proportion of persons engaged primarily in technical supervision increased rapidly with the age of respondents from 2 percent of those under 25 years of age to 32 percent of the group aged 35 to 39. The proportion remained fairly stable through age 54, then declined. (See chart 8 and appendix table A-11.) Similarly, the pro- portion of persons who were administrators increased up to about age 45, then became relatively stable, and finally declined. It is possible that the diminished proportions of super- visors and administrators in the oldest age groups reflect the operational status of mathematical employment prior to World War II, when most mathematicians worked as individual consultants rather than in groups. There was also some evidence that the pattern in the oldest age groups was influenced by the promotion to executive positions of the more experienced and capable supervisors. These persons, if they were no longer directly involved in mathematical work, usually were not considered within the scope of the survey. Employees whose major educational field was Chart 8. The majority of respondents 45-54 years of age were administrators or supervisors.... Percent in each age group in mathematical employment who were administrators, supervisors, or mathematical practitioners, 1960. 100 Mathematical 80 H. 2^ practitioners 60 H. Administrators. and 2" "N supervisors S. --~" "S. 40 F ./ e * / - /serior ./ 2 Y ~ J — - - - - ~ / ,’ ... • N. *se // & `- * “... 20 |- • / ...“ * - - .// ...“ .// .....” .Z.' ...“ Administrators | A .....…” O LL --"c...” | | | | | L | Under 25 30 35 40 45 50 55 60 to to to fo ło fo to fo 25 29 34 39 44 49 54 59 64 Age group Source: Table A-11. mathematics were less frequently found to be administrators or supervisors than those with majors in other subjects. This may be due to the relative youth and short experience of persons with mathematics degrees. Only one-third of the mathematics majors in the survey, but more than one-half of the respondents with college majors in fields other than mathematics, were 35 years of age or older. Educational Level. Excluding the respondents who had no degree, it appeared that the higher the educational level of respondents the greater the proportion engaged in administrative or super- visory work, when educational attainment and operational status were related without regard to other factors (table 13). However, the nondegree respondents who were administrators or super- visors were concentrated in the Department of the Army and in the insurance industry. When respondents from those two employers were subtracted, only 6.3 percent of the remaining employees without degrees were in administrative positions and 17.2 percent were in supervisory positions. - 19 TABLE 13–Operational status of persons in mathematical employment, by educational level, 1960 Percent distribution by operational status Number * -- Educational level reporting Total Adminis- Supervisor Mathematical trator practitioner All levels--------------------------------------- 18, 906 100 8. 3 20. 4 71. 3 Doctor's degree--------------------------------------- 670 100 9. 6 26.4 64. 0 Master's degree--------------------------------------- 2, 293 100 8. 0 24, 6 67. 4 Bachelor's degree------------------------------------- 5, 443 100 7. 1 17. 5 75. 4 No degree-------------------------------------------- 500 100 17. 4 22. 6 60. 0 1 Excludes 868 respondents who checked “other” for operational status and 208 others who did not Specify operational status or educational level. Although table 13 suggests that persons with advanced degrees are more likely to be adminis- trators or supervisors than are persons with only the bachelor’s degree, this relationship between education and operational status largely disappears when age is eliminated as a factor. Excluding respondents who have no degree, the proportion of respondents who were administrators or super- visors was approximately the same for each educational level within the same age group up to about age 50 (chart 9 and appendix table A-11). The high proportion of older respondents with bachelor's degrees who were administrators is again attributable to the large number of persons in this operational status who were in the insurance industry. Respondents from insurance companies comprised 47 percent of all holders of bachelor's degrees over 35 years of age who were adminis- trators. Many insurance company administrators and supervisors had become fellows or associates of a professional actuarial society by study equivalent to graduate-level work. In most industries other than insurance, the administrative and supervisory duties performed by employees with the doctorate very likely required a higher level of mathematical competence than that of the duties performed by those with lower educational attainment. Nevertheless, the data suggests that persons in mathematical employment who have only the bachelor's degree are about as likely to attain administrative or supervisory positions—at some level of difficulty—as are persons of the same age with advanced degrees. Organizational Unit. The proportion of em- ployees who are administrators or supervisors varies substantially by the type of operational unit to which they are attached. For example, supervisors and administrators together comprise only 13 percent of the employees in mathematics units but accounted for 45 percent of the persons in mathematical employment in production units (appendix table A-12). The data gathered by this survey do not explain differences in the ratio of supervisors and administrators to all employees for the various types of organizational units. How- ever, a preponderance of young employees and a relatively low proportion of supervisors and administrators again appeared as a pattern. Number of Employees Supervised. Survey re- spondents with supervisory duties were asked the total number of employees, in all kinds of work, under their direct or indirect supervision and, of this number, how many were professional person- nel engaged in primarily mathematical work requiring knowledge equal at least to that pro- vided by a 4-year college course with a major in mathematics. The respondents who indicated that they had supervisory duties exceeded the number of persons reporting their operational status as supervisor. This was to be expected since many administrators had employees under their supervision, and some persons who were primarily practitioners or were in some other operational status also supervised Some employees. All together more than one- third (35.5 percent) of all respondents stated that they had supervisory duties and, of this group, about 7 out of 10 Supervised employees engaged primarily in mathematical work. The majority of supervisors of mathematical personnel had 20 40 30 40 30 Bachelors \s 20 + degree S. | P^ N Master's degree 10 0 || - - Under 25 30 3.5 40 45 50 55 60 Chart 9. Within the same age group, approximately equal proportions at each educational level held administrative or supervisory positions.... Percent of persons in each age group in mathematical employment, who were administrators, supervisors, or mathematical practitioners, by education level, 1960 Administrators ſ | | | |2 -- No degree / \ º ..T. Bachelor's /* ſ degree ; | y #-cºs Doctor's- /.4% N. degree 22% \ | N H Master's degree Supervisors Doctors ^ …No degree 25 fo ło to to fo to fo to 29 34 39 44 49 54 59 64 Age group 100 tical T —I T practitioners Mathema \ Master's degree © \ ..." ©s \| | Doctors | degree T/ kºſ No degre © 40 N ^ / *—A- Bachelor's degree 30 2O 10 O Under 25 25 go 35 40 to fo to to 29 34 3 9 44 Age to to to 45 50 55 - 60 to 49 54 59 64 group Source: Table A-11. l 21 responsibility for fewer than five employees (see tabulation). Supervisors Number of employees Supervised Percent Number of all respondents All Supervisors--------------------- 3, 542 35. 5 Supervisors of employees engaged in primarily mathematical work- - - - - - 2, 550 25. 6 1-4 employees------------------ 1, 546 15. 5 5-9--------------------------- 555 . 5, 6 10-19------------------ — — — — — — — 265 2. 7 20-49------------------------- 140 1. 4 50-99------------------------- 32 ... 3 100 or more-------------------- 12 1 Mathematical Education Required for Current Position Employees’ opinions of the difficulty of their mathematical work were obtained by asking survey respondents to check the minimum mathe- matical education they believed necessary to perform the duties of their position, i.e., whether their work required less than a college degree or a bachelor's, master's, or doctor's degree with a major in mathematics." The information thus 7 Objective and detailed information on the mathe- matical content of respondents’ work was obtained by asking survey participants to check, from a list of 75 courses, the minimum mathematics courses required to perform the duties of their position. In addition, super- visors of persons in mathematical work were asked to check the minimum course requirements for a typical mathematics position at the bachelor's level under their supervision. The course data will be used by the Mathe- matical Association’s committee for this study to evaluate curriculum requirements for the teaching of applied mathematics. secured was compared with the actual educational attainment of respondents to determine the pro- portions who believed they were operating above, below, or at the educational level they had attained. Data on education attained and re- quired were also matched with years of experience of respondents to find out whether persons with long experience were more likely than those with short experience to feel that their educational background was being fully utilized. In addition, rough estimates were available on the extent to which employees used the education stated as the minimum required for their positions. Education Required Related to Attainment. One- fourth of the respondents reported that the educa- tion needed to perform the duties of their positions was less than a college degree with a major in mathematics. However, there is evidence that some respondents underestimated the level of mathematical education needed for their positions. Examination of a sample of replies indicated that many employees who stated that a major in mathematics was not needed for their positions had, in another part of the questionnaire, provided contradictory information. They had checked courses, as the minimum needed for their position, which would meet requirements for a mathematics major at many colleges and universities. Mathematics majors were more likely than respondents with majors in other fields to believe that a bachelor's or higher level degree in mathe- matics was necessary. About four-fifths of the mathematics majors, but only three-fifths of majors in other fields, believed that a college de- gree in mathematics was needed (table 14). Respondents who felt that less than a college TABLE 14.—Comparison of educational level attained and level of mathematical education believed required by respondent, for persons in mathematical employment, 1960 Percent distribution by educational level Number Major educational field of respondents reporting Total Doctor’s | Master’s | Bachelor’s | No degree degree degree degree All fields: Level attained--------------------------------------- 1 9, 774 100 7, 2 25. 7 61. 3 5. 8 Mathematics level required.--------------------------- 29, 755 100 4, 8 18. 0 51. 9 25. 3 Mathematics majors: Level attained-------------------------------------- 6, 559 100 6. 7 24, 1 65. 4. 3 3. 8 Mathematics level required.--------------------------- 6, 543 100 5. 7 18. 0 57. 6 18, 7 Other major fields: Level attained--------------------------------------- 3, 215 100 8. 4 29, 1 52. 9 3 9. 6 Mathematics level required.--------------------------- 3, 212 100 2. 8 18. 1 40. 2 38. 9 d Excludes 208 respondents who did not specify education major or level 8.55a]]]éOl. 2 Excludes 227 respondents who did not specify education major or level required. - ºr subject field classified according to field of greatest number of CreditS. 22 degree in mathematics was needed for their jobs reflect, to an unknown extent, a difference of opinion with their employers over the qualifica- tions for their positions. Employers were in- structed to distribute questionnaires only to persons in positions requiring knowledge equal at least to that provided by a 4-year college course with a major in mathematics.” A majority of the employees who had majored in mathematics believed that the educational levels they had formally attained matched the requirements for their positions. However, as shown in the tabulation below, 31 percent stated that the educational requirements for their posi- tions were below the level they had attained, and only 10 percent believed the requirements to be higher than their attained level. (See also ap- pendix table A-13.) Percent of mathematics majors citing educational level required for their positions aS— Educational level attained educational levels required for their positions were below their educational attainment: Same as at- Above at- || Below at- tained levelſtained levelſtained level All levels------------ 59. 5 10. 0 30. 5 Doctor's degree------------ 62.8 |-------- 37. 2 Master's degree------------ 40. 6 4. 1 55. 3 Bachelor's degree---------- 67. 7 10. 0 22. 3 No degree----------------- 34. 0 66. 0 |-------- Among degree holders with a major in mathe- matics, those with the bachelor's degree were most likely, and those with the master's degree least likely, to feel, that their educational back- grounds were being fully utilized. Two-thirds of the persons with no degree (but with college credits in mathematics) believed their positions required education equivalent to a college degree with a major in mathematics. The following tabulation indicates that mathe- matics majors who had less than 2 years of professional experience were more likely than persons with longer experience to believe that the * Some employers had difficulty in identifying employees who met the criteria and distributed questionnaires widely. This problem is discussed more fully in appendix B, page 67. 635761–62—5 “at least half the time” (table 15). Percent reporting educational level required for their posi- tion was below the level at- “ tained - Years of professional experience Bachelor’s Master’s degree degree holders holders All experience groups----- 22. 3 55. 3 Under 2 years------------------ 28, 1 77. 3 2----------------------------- 23. 0 62. 8 3----------------------------- 22, 4 60. 0 4----------------------------- 22. 0 60. 0 *- - - - - - dº º - - - - - sº - - - - - - - - - - - - - ºn E- 20. 2 64, 9 6-10-------------------------- 21. 5 52. 5 11-20------------------------- 18, 9 49. 5 21 or more--------------------- 16.4 50. 0 NOTE.-Holders of doctor’s degrees are not shown because few had less than 6 years of experience. Among bachelor's degree holders, the proportion reporting that the educational level required was below that attained continued to drop with length of experience. The striking impression of dis- satisfaction with the difficulty of the work assigned, in relation to educational attainment, was main- tained at all levels of experience by persons whose highest degree was the master's. Ectent of Use. The higher the level of mathe- matical education required for a position, the more fully the required mathematical ability was used. Approximately 8 out of 10 persons report- ing that their positions required mathematical education equal to a doctor’s degree stated that they used this training either “almost always” or On the other hand, as many as 6 out of 10 persons who believed performance of the duties of their positions did not require a college major in mathematics stated that they used their mathematics “less than half the time” or “almost never.” Two-thirds of all respondents indicating that they “almost never” used their mathematical education were those who reported that less than a college major in mathematics was needed for their positions. Considering all respondents, only 57 percent used the mathematical education cited as the minimum needed for their jobs half the time or more. 23 TABLE 15.-Eatent of use of minimum mathematical education cited by respondents as needed to perform the duties of their - - position, 1960 Level of mathematical education needed Extent of use All levels Doctor’s Master’s Bachelor’s No degree degree degree degree Number reporting-------------------------------------- 1 9,933 462 1, 764 5, 172 2,535 Percent distribution: - 9tal------------------------------------------- 100. 0 100. 0 100. 0 100. 0 100. 0 Almost always------------------------------------- 20, 8 42. 2 22. 7 20. 1 16. 9 At least half the time------------------------------- 35, 9 39. 8 45. 5 38. 5 23. 2 Less than half the time----------------------------- 37. 9 17. 1 30. 5 38.5 45. 7 Almost never-------------------------------------- 5.4 . 9 1. 3 2. 9 14. 2 1 Excludes 49 respondents who did not specify level of mathematical education required or extent of use. Major Functions in Mathematical Work Basic and applied research and development in the natural sciences and engineering were the major mathematical functions of more than half the survey respondents. About 6% times as many respondents were primarily concerned with applied research (45 percent) as with basic research (7 percent). Technical service allied to production was the only other specific function of primary concern to a large proportion (18 percent) of workers. The remaining functions about which respondents were queried and the percent checking each function are shown in table 16. Employees in mathematical work in private industry (excluding insurance) are most likely to be in applied research and development (52 per- cent) or in technical services allied to production (21 percent). (See chart 10 and appendix table A–14.) More than half the respondents in the following manufacturing industries were engaged in applied research and development: Transporta- tion equipment (except aircraft), aircraft and parts, electrical equipment, and professional and scientific instruments. On the other hand, only in the chemical and allied products industry were as many as 8 percent of the mathematical workers engaged in basic research. The proportion of employees in the manufacturing industries who were performing technical services allied to pro- duction ranged from about 14 percent in trans- portation equipment to 33 percent in petroleum products and extraction. As was true for other characteristics of insurance industry employees, the distribution by their chief function was unlike the pattern for other private industries. One-fourth indicated their function was administration. Other large num- major bers checked technical services, allied either to TABLE 16.-Major function of persons in mathematical employment, by type of employer, 1960 Type of employer tº Private industry Function All - Government Nonprofit employers Industry Organization Insurance excluding 1I]SUIT 3.11C0. Number reporting-------------------------------------- 1 9, 867 914 6, 189 2, 542 222 Percent distribution: - Total------------------------------------------- 100. 0 "100. 0 100. 0 100. 0 100. 0 Basic research in the natural sciences and engineering--- 7. 0 . 1 5. 5 11. 4 25. 7 Applied research and development in the natural sciences - and engineering---------------------------------- 45.4 2. 9 51. 8 44, 8 50. 4 Nontechnological research, including marketing and • * . . other economic research---------------- — — — — — — - - - - - 4. 0 5. 0 3. 8 3. 6 10. 4 Technical Services allied to production----------______ 18, 3 19. 6 21.4 11. 7 2. 2 Technical Services allied to Sales, promotion, or distri- bution------------------------------------------ 4. 4 17. 4 4, 1 ... 8 ... 5 Teaching and training------------------------------ ... 8 ... 3 . 9 ... 8 . 9 Administration------------------------------------- 8. 2 25. 5 4, 6. 11. 1 2. 2 Other--------------------------------------------- 11. 9 29. 2 7. 9 15. 8 7.7 1 Excludes 115 respondents who did not specify function. (24 Chart 10. Research was the chief function of mathematical workers in most fields of employment.... Tercent distribution of persons in mathematical employment, by major function, in major fields of employment, 1960 O 20 40 60 80 106 I I T -I Nonprofit organizations / / / / / / / / zzz z / / / / / / / / / / / / , z. z. z / / / º Government Insurance z z z / / / / / / / / / / / / / / / / / / / / / / / / / / / 2 < z z z z < z z < z < z z. z zzzzzzzzzz z z z z z 2 & 2 × < z_2 z z Z Z Z Z ZZ Z & Z Z_2 < * Z * Z ~ * Z.Z Z.Z. ga Basic and applied research in the natural sciences and engineering. º 2 Other Technical services Source: fable 16. production or to sales, promotion, or distribution. Only 3 percent were concerned with research and development in the natural sciences and engineer- ing. It was apparent that many respondents from the insurance industry felt that none of the classifications provided on the questionnaire was suited to their functions, and 29 percent checked “other” when answering this item. Basic research in the natural sciences and engineering was the chief function of a much larger proportion of Federal Government em- # ployees (11 percent) than of private industry em- ployees (5 percent). Much of the Government basic research work was accounted for by the National Aeronautics and Space Administration. Of NASA respondents, 45 percent indicated that they were primarily engaged in basic research. Nevertheless, when these employees were sub- tracted from the total, the share of Government employees engaged in basic research was still about 8 percent. The higher proportion of Department of Commerce employees in basic research in the natural sciences and engineering largely reflects the type of activity carried on by that Department's Bureau of Standards. About 45 percent of all Government employees were engaged in applied research. Among employees of the Department of the Navy—an agency which operates a number of laboratories for the design and development of weapons, electronic equip- ment, and ships—three-fifths reported that they were engaged chiefly in applied research and development in the natural sciences and engineer- ing. Technical services allied to production was the primary function of about one-eighth of the respondents from the military agencies and the Department of Commerce. Although a higher proportion of Government than of industry respondents was engaged in administration, this was entirely due to the large proportion of adminis- trators found among Department of Army employees. (See related discussion on p. 19.) Since most of the respondents from nonprofit organizations were employed by research centers or institutes, research was the major function of three-fourths of them; 26 percent were in basic research and 50 percent in applied research in the natural sciences and engineering. 25 INCOME FROM MATHEMATICAL EMPLOYMENT, 1960 Average (median) income from the major posi- tions of persons in mathematical employment other than teaching was $8,500 in 1960. The middle 50 percent of the incomes ranged from about $7,000 to $11,000. Only 2 percent of the respondents earned less than $5,000, and 7 percent earned $15,000 or more (table 17).' Median in- comes were highest for groups with advanced de- grees, for those in the older (but not oldest) age brackets, and for supervisors, men employees, and workers in private industry, especially insurance. Educational Level, Age, and Sex In general, the higher the educational attain- ment of respondents, the higher their earnings. Income increased with age and experience through- out most of the working lives of the respondents, but average incomes of the oldest respondents at each educational level were lower than for some of the younger age groups. Women had lower incomes, on the average, than men in the same education and age groups (chart 11). Median annual income from the major positions of men with doctor's degrees was $13,100–30 per- cent above the median for men with master's degrees and about 60 percent higher than for those with the bachelor's. Incomes of women were also markedly higher for those with graduate degrees (table 18). In most age groups where data are comparable, median income for men with doc- torates was $2,000 to $3,000 higher than for men with master's degrees (chart 11 and appendix table A-15). Median incomes of men with mas- ter's degrees were $1,000 to $1,500 higher in each age group, below age 45, than the incomes of men with the bachelor's. Men who had not earned a degree (almost all had earned college credits) had median incomes consistent with the pattern of lower educational level—lower income in all age groups for which medians could be computed. The relatively low earnings reported by women employees repeats similar patterns found in other studies which discuss the employment of women 1 Persons filling out the survey questionnaire were not asked to state their exact income but were given a list of brackets to check (see facsimile of questionnaire in ap- pendix C). Although the income question was marked “optional,” about 95 percent of the respondents supplied answers to this question. TABLE 17.-Distribution of persons in mathematical emplº by annual income from major position and by educational level, 1960 Highest educational level Annual income All Doctor’s Master’s Bachelor’s No levels degree degree degree degree Number reporting------- ------------------------------- 19, 284 656 2, 362 5, 717 549 Percent distribution: otal------------------------------------------- 100. 0 100. 0 100. 0 100. 0 100. 0 Under $4,000-------------------------------------- - 1 ---------- . 1 ... 2 ... 2 $4,000-$4,999–------------------------------------- 1. 9 ... 2 ... 3 2. 7 2. 9 $5,000–$5,999–-------------. ----------------------- 8. 6 ... 2 1. 7 12. 3 9. 3 $6,000-$6,999---------- - -------------------------- 16. 0 . 5 5. 7 21. 9 16. 6 $7,000-$7,999–------------------------------------- 16. 3 1. 2 14, 1 18, 2 } 24, 5 $8,000-$8,999–------------------------------------- 13. 5 3. 2 15. 4 13. 4 18. 5 $9,000-$9,999–------------------------------------- 10. 0 5. 8 13. 8 9. 3 6. 2 $10,000–$10,999–--------- - - - - - - - - - - - - - - - - - - - - - - - - - - 9. 2 11. 3 13. 5 7. 3 7. 5 $11,000-$11,999------------------------------------ 7. 2 15. 5 11. 4 -4. 7 4. 7 $12,000-$14,999------------------------------------ 10. 5 37. 3 15. 8 5. 8 4, 6 $15,000–$19,999_____ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 4. 5 19. 8 6. 0 2.4 2. 6 $20,000 and over----------------------------------- 2, 2 5. 0 2. 2 1. 8 2. 4 ! Excludes 531 respondents who did not specify income and 167 who did not specify educational level (including 9 who failed to specify both income and educational level). 27 Chart 11. Median incomes were generally higher for persons with advanced degrees.... º: Median incomes of men and women in mathematical 16 employment, by age and educational level, 1960 | P. egree - 14 \ t Hº: - - * *'. | Master's ,’’’ / º, degree ,” Bachelor's ** 12 L-zº -degree — X-MEN 2*TTA, Nº NS 2 ...” degree 10. <+--!>< * X-- - - yz ~ 21 ſº ,” ... --" / * Master's/ — 23–2 d * - - 8 2 .”[2" | Dº" WOMEN , -’ I - -º-TT Bachelor's/ 6 2’ - degree 4 2 O zk Under 25 30 35 40 45 50. 55 60 25 to to to ło fo fo to to 29 34 39 44 49 54 59 64 Age group Source: Table A-15. in professional occupations.” The incomes of women just beginning work were only moderately lower than men's starting incomes, but the gap widened in the older age groups. Median income for women under 30 years of age, with bachelor's degrees, was only 9 percent below that for men in the same age and education group, but at age group 45 to 49 this differential had risen to 32 percent. A comparison of bachelor's degree holders in the age groups under 25 years and 45 TABLE 18.-Median annual incomes of men and women in mathematical employment, by educational level, 1960 All respond- Men ents Educational level Women All educational levels- $8,500 Doctor's degree------------| 13,000 $8,900 $6,600 13, 100 11, 000 Master's degree------------ 9,900 | 10, 100 8, 000 Bachelor's degree---------- 7, 700 8, 100 6, 500 No degree----------------- 7, 900 8,000 6, 400 * For a comparison of incomes of men and women chemists, see Manpower Resources in Chemistry and Chemical Engineering, Bureau of Labor Statistics Bull. 1132, 1953, p. 36; for comparisons of social scientists, see Personnel Resources in the Social Sciences and Humanities, Bureau of Labor Statistics Bull. 1169, 1954, p. 118. to 49 years shows that median income of men was 75 percent higher for the older age group; for women, it was only 32 percent higher. (See appendix table A-15.) This strongly suggests that, compared with men, women had fewer opportunities for professional advancement. Pos- sible explanations of the income differences are that women were not chosen as frequently as men for responsible positions because marriage or motherhood might result in their resignation; some women may have left the labor market for a period because of marriage and family responsi- bilities and because they did not have as many years of continuous experience as men in the same age group; and women may have done less job hunting than men, and if married they probably limited their job hunting to the areas in which their husbands worked. Incomes of men increased fairly rapidly into middle age, then the growth slowed. For both men and women, the oldest age group at each educational level had an average income below peak.” The early peak reached at age 40 to 44 by men with doctor's degrees is explained, in part, by the high proportion of older persons with that degree who were employed in Government, where salaries were lower, on the average, than in private industry. When examined separately, it was found that in Government and in private industry excluding insurance, average salaries of persons with the doctorate did not decline until after age 55. The relatively sharp climb in income after age 49 for men with bachelor's degrees is related to the number of older respondents with this educational attainment who were employed in the insurance industry, where incomes were relatively high. Respondents from the insurance industry accounted for only 14 percent of all men respondents with bachelor's degrees but comprised 37 percent of those aged 50 years or older. Income patterns for the insurance industry, as well as for other types of employers, are discussed in the section which follows. Type of Employer Incomes from mathematical employment in the insurance industry were, on the average, much * Herman P. Miller, analyzing data collected by the Bureau of the Census, reported that in 1958 maximum earnings of high school and college graduates were found in the 45 to 54 age group. See “Annual and Lifetime Income in Relation to Education, 1939–59,” American Economic Review, December 1960, p. 973. 28 TABLE 19.-Median annual income of persons in mahºº! employment, by princpal type of employer and educational evel, 1960 Highest educational level Employer - . All Doctor’s Master’s Bachelor’s NO levels degree degree degree degree All employers------------------------------------ $8,500 || $13,000 $9,900 $7,700 $7,900 Private industry------------------------ --------------- 8, 800 13, 300 10, 100 7, 900 7, 800 Insurance industry--------------------------------- 10, 900 1. 13,000 10, 000 13, 500 Private industry excluding insurance----------------- 8, 700 13, 300 10,000 7, 700 7, 600 Federal Government----------------------------------- 7, 900 11, 800 9, 100 7, 300 7,900 Nonprofit organization---------------------------------- 9, 400 14, 200 9, 600 6,900 (i) 1 Too few cases to compute median. higher than those received from employment in other segments of private industry, in the Federal Government, or in nonprofit organizations. Me- dian incomes were higher in private industry than in Government for persons at all degree levels, but not for those without degrees. However, the highest average income for any group was that reported by employees with the doctorate who were working for nonprofit organizations. (See table 19 and appendix table A-16.) * In the insurance industry the outstanding feature of the income pattern was the proportion of employees with high incomes regardless of formal educational attainment. Almost 30 per- cent had annual incomes of $15,000 or more a year, and 15 percent had incomes of $20,000 or more. (See appendix table A-18.) As mentioned earlier in this report, many of the respondents from the insurance industry were associates or fellows of a professional actuarial Society, and attaining this status in that industry is undoubtedly more sig- nificant in determining income than is obtaining a college degree. More than 70 percent of the re- spondents from insurance companies indicated they were concerned with actuarial work, and 9 percent were vice presidents or assistant vice presidents of their companies. Average incomes of employees with bachelor's degrees in industries other than insurance were relatively close to one another at each age (and experience) level. Incomes in private industry, excluding insurance, were appreciably higher on the average than the income of Government employees for persons at each degree level, partic- ularly in the older age groups. No formal limit * Median incomes by educational level and age are also shown for specific types of operational units. (See appendix table A-17.) exists for salaries in private industry. Government civil service employees, on the other hand, had a legal maximum on their salaries—$17,500 up to July 1, 1960, and $18,500 after that date,” except for a few persons in special positions. Less than 1 percent of Government employees, but nearly 6 percent of private industry employees (excluding insurance) reported annual incomes of $15,000 or more. (See chart 12 and appendix table A-18.) * Supervisors and Nonsupervisors Median incomes of supervisors were about a third higher than for nonsupervisors in Govern- ment employment, not quite two-fifths higher in private industry excluding insurance, and three- fourths higher in the insurance industry. How- ever, a larger share of supervisors than of non- supervisors were in the older age and higher education groups in which incomes generally are higher, and the income difference was not as large at most specific age and education levels as for the total. In most of the age and education groups, in both manufacturing and Government, the differ- * About 5 percent of the response from Government employees was mailed after July 1, 1960. ° Other recent surveys provide data on earnings of math- ematical employees in private industry and Govermnent. A survey made by the American Mathematical Society of starting salaries of persons with the Ph.D. in mathematics (the majority of whom had at least 1 year of work experi- ence) showed median starting salaries in the spring of 1960 of $11,000 in industry, $9,300 in Government, and $6,500 in university and college teaching. See “Starting Salaries for Mathematicians with a Ph. D.,” American Mathema- tical Society Notices, October 1960, pp. 598–99. For infor- mation on a survey of salaries and cash bonuses of math- ematicians in positions of seven specifically defined levels of difficulty and responsibility in private industry, see National Survey of Professional, Administrative, Technical and Clerical Pay, Winter 1959–60, Bureau of Labor Statistics, Bull. 1286, October 1960. 29 Chart 12. Incomes were highest in private industry.... Median and quartile incomes of persons in mathematical employment in government and in private industry (excluding insurance), by age and educational level, 1960 Thousands of dollars 20 16 12 |Doctor's degrees - 3d quartile —- PRIVATE e e e º 'º º T . . . . . . . . . . . . . . * * * * * * * ... • . . . * * * * * * INDUSTRY ‘,- ~ * T-P- ~rl.... GOVERNMENT e e e s e º e 2: : \ --- MEDIAN –- © . . . . . " • , , , , , a e s - e < * * * * * * * * * * * * * * * * * 1st quartile—— ... • * * * * * * * : MEDIAN 1st quartile j |Master's degrees 25 30 35 40 45 to fo ło to fo 29 34 39 44 49 Age group. -i. 50 55 to fo 54 59 Source: Table A-16. 30 TABLE 20.-Amount and percent by which median annual income of Supervisors eacceeded income of nonsupervisors in private industry and Government, by educational level and selected age groups, 1960 ! Private industry Government Age group (years, at nearest birthday) Private industry excluding Insurance IITSUIT811C0 Amount Percent Amount Percent Amount Percent Alleducationalieves All age groups------------------------------- $3,000 38 $5,700 74 $2,300 32 Bachelor's degree All age groups-------------------------------- $2,600 36 $5,400 74 $2, 200 33 25-29 years--------------------------------- 1, 300 19 1, 000 15 1, 400 22 30-34--------------------------------------- 1, 600 20 1, 500 16 1, 400 I9 35-39--------------------------------------- 1, 800 18 -------------------- 1, 700 22 40-44--------------------------------------- 1, 500 16 -------------------- 2, 300 30 45-49--------------------------------------- 3, 100 35 -------------------- 1, 500 18 50-54--------------------------------------- 1, 100 11 ----------|---------- 1, 900 26 Master’s degree All age groups------------------------------- $2,400 26 $6,000 67 $2,300 28 25-29 years--------------------------------- 1, 300 16 ----------|----------|----------|---------- 30-34--------------------------------------- 1, 500 16 -------------------- 2, 500 31 35-39--------------------------------------- 1,600 15 -------------------- 1, 700 19 40-44--------------------------------------- 3,000 29 -------------------- 1, 600 17 45-49--------------------------------------- 2,500 28 ----------|---------- 1, 300 14 50-54-----------------------------------------------------------|----------|---------- 1, 700 20 Doctor’s degree All age groups------------------------------- $2,600 21 ----------|---------- $700 6 30-34 years--------------------------------- 1, 400 12 ----------|------------------------------ 35-39--------------------------------------- 1, 800 13 ---------------------------------------- 40-44--------------------------------------- 2, 100 16 ----------|----------|----------|---------- 1 Comparisons are not shown for cells with fewer than 20 respondents. ences between the median incomes for supervisors for persons with advanced degrees. Half the and nonsupervisors were from about $1,300 to $1,800 per year. In about half the age groups, earnings of supervisors exceeded those of non- supervisors by from 15 to 20 percent. The smallest percentage difference occurred at the highest educational level. The amounts and percentages by which supervisor's median in- comes exceeded the incomes of nonsupervisors are shown in table 20; the actual medians are shown in appendix table A-19. Additional Professional Income About 1 in 7 of the respondents had professional income from a source outside his major position." The proportion with additional income was highest 635761–62—6 incomes earned from outside professional work were under $500 a year. One-third of the persons with doctorates earned outside income, including 14 percent who earned $1,000 or more a year. On the other hand, only 2 percent of those with bachelor’s degrees and 4 percent with no degree earned $1,000 or more a year in outside incomes (table 21). As reported earlier (page 10), the most common source of outside income was teaching at colleges and universities. 7 Additional income was defined as professional income earned during the preceding year from a source other than major current position, specifically excluding nonpro- fessional income such as investments. 31 TABLE 21–Percent of persons in mathematical employment with income from professional work other than their major position, by amount of additional income at each educational level, 1960 - ~. Highest eductional level Additional income All levels Doctor’s Master’s Bachelor’s No degree degree degree degree All respondents---------------------------------- 100. 0 100. 0 100. 0 100, 0 100. 0 No additional income----------------------------------- 85. 3 66. 6 79. 0 90. 1 86. 0 With additional income--------------------------------- 14. 7 33. 4 21. 0 9. 9 14. 0 $1-$499------------------------------------------- 7.2 10. 5 8. 9 5. 9 8. 0 $500−$999----------------------------------------- 3. 2 8. 8 5. 3 1. 8 1. 7 $1,000-$1,499-------------------------------------- 2. 2 7. 9 3. 3 1. 1 2. 1 $1,500−$1,999-------------------------------------- ... 8 1. 7 1. 4 . 5 ... 2 $2,000-$2,999–------------------------------------ ... 6 1. 8 1. 2 ... 2 ... 7 $3,000-$3,999–------------------------------------- ... 3 1. 1 ... 4 ... 1 ... 2 $4,000-$4,999–------------------- - - - - - - - - - - - - - - - - - - ... 1 ... 6 . 1 . 1 ... 2 $5,000 and over------------------------------------ ... 3 1. 0 ... 4 ... 2 . 9 32 Appendix A STATISTICAL TABLES Tables in this section contain data which support the analyses presented in the main part of the report. In addition, some tables are included which cover items to be analyzed in a separate report by the Survey Committee of the Mathematical Association of America. The additional data prepared for the Committee (included in tables A–20 through A-35) are con- cerned largely with the mathematical content of the positions filled by respondents and with curriculum requirements in the field of applied mathematics. 33 TABLE A-1.-Educational attainment of men and women in mathematical employment, 1960 Total Men Women Educational level Number Percent Number Percent Number Percent All levels----------------------------- 1 9, 815 100. 0 8, 336 100.0 1,394 100. 0 Bachelor's degree---------------------------- 6,018 61. 3 4, 866 58.4 1,099 78. 8 Master's degree------------------------------- 2, 519 25. 7 2, 296 27. 5 203 14, 6 Doctor's degree------------------------------ 712 7.2 682 8. 2 25 1. 8 No degree----------------------------------- 566 5. 8 492 5. 9 67 4, 8 1 Excludes 167 persons not reporting education. Components may not add to the totals because some individuals did not specify their sex. TABLE A-2.-Age, sex, and educational attainment of persons in mathematical employment, 1960 All educational levels Doctor's degree Master’s degree Age group (years, nearest birthday) Total Men Women Total Men Women Total Men Women Number reporting------------------------ 19, 718 8, 307 | 1, 384 697 669 25 2, 453 2, 247 199 Percent distribution: Total----------------------------- 100. 0 || 100. 0 || 100. 0 || 100. 0 || 100. 0 || 100. 0 || 100. 0 || 100. 0 || 100. 0. Under 25 years----------------------- 9. 1 6. 3 25, 5 --------------|------- 2.6 2.4 5. 5 5–29------------------------------- 26. 0 25, 5 28. 8 9. O 9. 3 4. 0 20. 8 20. 2 26. 7 30-34------------------------------- 25. 2 27, 1 13. 4 26, 4 26. 6 24. 0 27, 7 28, 8 17. 6 35-39------------------------------- 17. 3 18. 3 11. 6 29. 0 29. 7 16. 0 23. 1 23. 7 16. 6 40-44------------------------------- 9. 5 9. 8 8. 1 13. 5 12. 7 24. 0 12.4 12. 6 10. 1 45-49------------------------------- 5. 9 6. 1 5. 3 10. 0 10. 0 12. 0 6. 4 6, 2 8. 0 50-54------------------------------- 4. 0 . 3. 9 4. 4 6. 5 6. 3 12. 0 4, 2 3. 7 8. 5 55-59------------------------------- 1. 9 1. 9 1. 8 2. 9 2. 8 4. 0 1. 9 1. 7 4. 5 60-64------------------------------- ... 8 ... 8 ... 7 1. 7 1. 6 4. 0 ... 6 . 5 1. 0 65 and over-------------------------- ... 3 ... 3 4 1. 0 1.0 ------- 3 ... 2 1. 5 Median age------------------------------ 32. 5 32. 8 28, 7 37. 0 36. 9 40. 7 34. 3 34.3 34, 6 Bachelor’s degree No degree Level not specified Total Men Women Total Men Women Total Men Women Number reporting------------------------ 5, 864 || 4, 773 | 1,077 551 482 66 153 136 17 Percent distribution: Total----------------------------- 100. 0 || 100. 0 || 100. 0 || 100. 0 || 100. 0 || 100. 0 || 100. 0 || 100. 0 || 100. 0 Under 25 years----------------------- 13. 2 9. 3 30. 8 6. 9 5. 8 13. 6 2. 6 1. 5 11. 8 25–29------------------------------- 31. 5 31. 6 30. 6 16, 9 16. 4 18. 2 10, 5 9. 6 17. 6 30-34------------------------------- 24, 8 27. 5 12.4 18, 9 19. 7 15. 2 16, 3 17. 6 5. 9 35-39------------------------------- 13. 4 14, 2 9. 7 19. 1 18. 9 21. 3 15. 7 14. 7 23. 6 40-44------------------------------- 7. 4 7. 5 7. 1 12. 7 13. 1 10. 6 15. 0 14. 7 17. 6 45-49------------------------------- 4, 3 4. 4 4. 1 13. 6 13. 9 10. 6 15. 0 14. 7 17. 6 50-54------------------------------- 3. 1 3.0 3. 7 5. 8 6. 4 1. 5 16. 3 18, 4 |_ _ _ _ _ _ 55-59------------------------------- 1. 5 1. 6 1. 0 3. 4 3. 3 4. 5 5. 9 5. 9 5. 9 60-64------------------------------- ... 6 ... 6 . 5 2. 2 2. 3 1. 5 ... 7 7 ------ 65 and over-------------------------- ... 2 ... 3 . 1 . 5 ... 2 3. 0 2. 0 2. 2 ------ Median age------------------------------ 30. 6 31. 1 27. 6 36. 5 36. 6 35. 2 41. 1 41. 7 37. 6 "Excludes 264 persons not reporting age. Components may not add to totals because some individuals did not specify their Sex. 34 TABLE A–3.-Educational level and major subject field of persons in mathematical employment, 1960 Percent distribution by educational level Number Major subject field reporting Total Doctor’s | Master’s Bachelor’s | No degree k degree degree degree All major subject fields------------------------------ 29, 815 100 7. 2 25. 7 61. 3 5, 8. Mathematics------------------- - - - - - - - - - - - - - - - - - - - - - - - - - 6, 559 100 6, 7 24. 1 65. 4 3. 8. Mathematics---------------------------------------- 6,096 100 6. 3 22. 2 68. 0 3. 5. Statistics------------------------------------------- 369 100 16. 0 50. 1 29. 0 4. 9. Actuarial science------------------------------------- 94 100 || ------- 40. 4 41. 5 18, 1 Physical science----------------------------------------- 7.59 100 17. 1 24, 9 53. 8 4. 2. Physics--------------------------------------------- 504 100 | 16. 3 || 27. 0 || 52. 7 4. 0. Chemistry------------------------------------------ 148 100 23. 6 12. 8 59. 5 4. 1 Earth sciences--------------------------------------- 52 100 15. 4 32. 7 50. 0 1. 9. Other---------------------------------------------- 55 100 9. 1 30. 9 50. 9 9. 1 Engineering--------------------------------------------- 1, 448 100 5. 6 29. 4 57. 0 8. 0. Electrical and electronic------------------------------ 411 100 3. 4 31.4 58. 9 6. 3 Mechanical----------------------------------------- 369 100 5. 7 26. 8 60. 7 6, 8, Chemical------------------------------------------- 178 100 12. 9 21. 3 60. 7 5. 1 Aeronautical---------------------------------------- 128 100 5. 5 43. 8 48. 4 2. 3 Industrial------------------------------------------- 84 100 4. 8 44. 0 42. 9 8. 3 Civil----------------------------------------------- 80 100 1. 2 20. 0 70. 0 8, 8. Other---------------------------------------------- 198 100 5. 6 25. 2 49. 5 19. 7 Business and commerce----------------------------------- 350 100 ... 6 31.4 44. 0 24. 0. Social sciences------------------- ------------------------ . 215 100 7. 4 21. 9 64. 2 6. 5, Education---------------------------------------------- 128 100 1. 6 64, 8 26. 6 7. 0. Humanities--------------------------------------------- 91 100 6. 6 14. 3 60. 4 18. 7 Biological, medical, and agricultural science----------------- 71 100 23. 9 16, 9 49. 2 10. 0. Psychology--------------------------------------------- 64 100 17. 2 32. 8 42. 2 7. 8, Other and not specified----------------------------------- 130 100 4. 6 30. 0 39. 2 26. 2. 1 Major subject field classified according to field of greatest number of college credits. 2 Excludes 167 respondents who did not specify educational level attained. TABLE A-4.—Shifts in major subject field between baccalaureate and advanced degree, for persons in mathematical employ- ment, 1960 Persons with bachelor’s degree in mathematics and advanced degree in Persons with bachelor's degree in another field and advanced degree in another field mathematics Field of advanced degree Number Field of bachelor's degree Number Total.--------------------------------- 285 Total-------------------------------- 504 Physics-------------------------------------- 58 || Physics------------------------------------- 96, Other physical sciences------------------------ 19 || Chemistry---------------------------------- 42. Electrical engineering------------------------- 18 || Other physical Sciences----------------------- 18. Other engineering----------------------------- 37 || Electrical engineering------------------------ 52. Pducation----------------------------------- 57 || Chemical engineering------------------------ 37 Business and commerce------------------------ 35 || Mechanical engineering----------------------- 36 Social sciences-------------------------------- 17 || Other engineering---------------------------- 42. Psychology---------------------------------- 11 || Social sciences------------------------------- 36 Humanities---------------------------------- 9 || Humanities--------------------------------- 22. All other fields------------------------------- 24 || Business and commerce----------------- - - - - - 21 Education---------------------------------- 17 All other fields------------------------------ 85. 35, TABLE A-5.-Age at receipt of highest degree of persons in mathematical employment, 1960 Highest educational level attained Age at receipt of highest degree (years at nearest birthday) Doctor’s Master’s Bachelor’s Doctor’s Master’s Bachelor’s degree degree degree degree degree degree All major subject fields 1 Mathematics majors 2 Number reporting--------------------------- 695 2, 436 5, 824 430 1, 532 4, 151 Percent distribution: Total-------------------------------- 100. 0 100. 0 100. 0 100. 0 100. 0 100. 0 Under 20 years------------------------- . 1 (8) . 9 ... 2 . 1 ... 8 *-------------------------------------|---------- ... 3 2, 6 ||---------- . 5 2. 7 *------------------------------------- . 1 1, 2 14-7 ||---------- 1. 0 15. 3 *------------------------------------- . 9 5. 3 29. 7 1. 4 5. 7 32. 6 *------------------------------------- 1. 2 11. 5 13. 8 1. 2 11. 9 13. 3 *------------------------------------- 5. 0 12. 3 8. 5 4. 9 11. 9 7. 5 *------------------------------------- 8. 3 12. 1 7. 9 8. 4 12. 4 7. 4 *------------------------------------- 9. 8 11. 2 6. 2 8. 8 12. 1 6. 2 *7------------------------------------- 11. 6 9. 4 4, 9 9. 5 9. 4 4, 8 *------------------------------------- 11. 1 7, 7 2. 9 11. 4 7. 9 2. 8 *------------------------------------- 9. 9 6, 6 2. 3 8. 1 6, 7 2. 2 *------------------------------------- 8, 2 5. 0 1. 4 7. 9 4. 1 1. 2 *------------------------------------- 7. 8 4, 1 1. 1 8. 8 4, 1 ... 8 *------------------------------------- 5. 8 3. 0 ... 7 7. 0 2, 6 . 5 *------------------------------------- 3. 7 2. 0 . 5 4. 7 1. 6 . 4 *------------------------------------- 2. 7 1. 4 ... 4 2. 8 1. 4 ... 3 *~39---------------------------------- 9, 2 4, 8 1. 0 9. 3 4, 8 . 9 49-44---------------------------------- 3. 9 1. 4 ... 4 4, 7 1. 1 ... 2 45-49---------------------------------- . 4 ... 6 ... 1 ... 7 ... 6 . 1 50 and over---------------------------- ... 3 . 1 (8) 2 1 ---------- Median age at receipt of highest degree-------- 28. 7 26. 2 22. 6 29. 0 26. 0 22. 5 *Excludes 294 respondents who did not specify the age at which they received their highest degree—17 at the doctoral level, 83 at the master's level, and 194 at the bachelor’s level. * Excludes 198 respondents who did not specify the age at which they received their highest degree—11 at the doctoral level, 48 at the master’s level, and 139 at the bachelor’s level. & Less than 0.05 percent. TABLE A-6.-Age at receipt of bachelor’s degree of persons in mathematical employment, 1960 All major subject fields 1 Mathematics majors 2 Age at receipt of baº. degree (years at nearest Highest educational level attained Highest educational level attained ITth Ola, y) All levels All levels Doctor’s Master's Bachelor’s Doctor’s Master’s Bachelor’s. degree degree degree degree degree degree Number reporting---------------------- 8, 862 670 2, 368 5, 824 || 6,063 417 | 1,495 || 4, 151 Percent distribution: otal.--------------------------- 100. 0 100. 0 100. 0 100. 0 100. 0 100. 0 100. 0 100. 0 Under 20 years--------------------- 1. 2 3. 4 1. 2 . 9 1. 2 3. 8 1. 3 ... 8 20-------------------------------- 4. 1 14. 0 5. 1 2, 6 4. 2. 16. 3 4. 9 2. 7 *-------------------------------- 16. 3 20. 4 19. 0 14, 7 16. 7 19, 2 19. 9 15. 3 *-------------------------------- 29. 5 26. 3 29. 7 29. 7 31.4 24, 1 31. 0 32. 6 *-------------------------------- 13. 5 12.4 13. 2 13. 8 13. 1 12. 9 12. 5 13. 3 *-------------------------------- 9. 0 9. 0 10. 3 8, 5 8. 3 9. 6 10. 1 7. 5 *-------------------------------- 7. 4 6. 6 6. 6 7. 9 7. 2 5. 5 7. 0 7.4 20-------------------------------- 5. 6 2. 7 4. 8 6, 2 5. 5 2. 2 4. 3 6. 2 27-------------------------------- 4. 1 2. 5 2. 6 4. 9 4. 1 3. 6 2. 2 4, 8 *-------------------------------- 2. 7 . 9 2. 6 2. 9 2. 6 1, 2 2. 6 2. 8 *-------------------------------- 1. 9 ... 3 1. 3 2. 3 1. 8 ... 5 1. 2 2, 2 30-------------------------------- 1. 1 ... 3 7 1. 4 9 -------- ... 3 1. 2 31 and over------------------------ 3. 6 1. 2 2. 9 4, 2 3. 0 1. 1 2. 7 3. 2 Median age at receipt of bachelor's degree- 22. 5 22. 0 22. 3 22. 6 22.4 21. 9 22. 3 22. 5 * Excludes 387 respondents who did not specify the age at which they re- 3 Excludes 248 respondents who did not specify the age at which they re- ceived their bachelor’s degree—42 of whom had attained the doctorate, 151 Who attained master's degree, and 194 whose highest level was the bachelor’s degree. ceived their bachelor's degree—24 of whom had attained the doctorate,85 who had attained the master's degree and 139 whose highest level was the bachelor’s degree. 36 TABLE A-7.-Years of professional eacperience of persons in mathematical employment, 1960 Replies Replies Years of experience 1 Years of experience Number Percent Number Percent All experience groups---------- 29, 876 100. 0 Less than 6 months.----------------- 129 1. 3 || 14-15------------------------------ 361 3. 7 1 year----------------------------- 823 8. 3 || 16-17------------------------------ 295 3. 0 2---------------------------------- 780 7. 9 || 18-19------------------------------ 286 2. 9 3---------------------------------- 854 8. 7 || 20–21------------------------------ 212 2. 1 4---------------------------------- 787 8.0 || 22-23------------------------------ 187 1. 9 5---------------------------------- 687 7. 0 || 24–25------------------------------ 173 1. 8 6---------------------------------- 508 5. 1 || 26-27------------------------------ 117 1. 2 7---------------------------------- 481 4. 9 || 28–29------------------------------ 81 ... 8 8---------------------------------- 584 5. 9 || 30-31------------------------------ 91 . 9 9---------------------------------- 669 6. 8 || 32–33------------------------------ 74 ... 7 10--------------------------------- 645 6, 5 || 34–35------------------------------ 61 ... 6 11--------------------------------- 348 3. 5 || 36-37------------------------------ 31 ... 3 12--------------------------------- 328 3. 3 || 38–39------------------------------ 22 ... 2 18--------------------------------- 223 2, 3 || 40 and over------------------------ 39 ... 4 1 Data in this table do not correspond to data in table A-8 because of differ- ences in rounding. For this table, fractional years were rounded to the nearest whole year, i.e., 1 year and 5 months was rounded to 1 year; 1 year and 7 months was rounded to 2 years. Since the bulk of the replies were received in mid-1960, a respondent included in the 1 year of experience group in this table is assumed to have started work in calendar year 1959. * Excludes 106 respondents who did not report years of experience. TABLE A–8.—Number of employers in last 10 years 1 for persons in mathematical employment, by eacperience level, 1960 lower whole year except for replies showing less than 1 full year’s experience, Percent distribution by number of employers Years of experience 2 Number reporting Total 1 2 3 4 5 Or more All experience levels----------------------- 39, 876 100 52, 9 23. 6 13. 0 6. 2 4. 3 Under 3 years---------------------------------- 1, 862 I00 76. 5 18. 2 3. 5 1. 1 ... 7 3-5-------------------------------------------- 2, 251 100 52. 7 28. 1 12. 8 4. 4 2. 0 6-10------------------------------------------- 2, 869 100 37. 0 26. 6 19. 1 10. 1 7. 2 11-15------------------------------------------ 1, 235 100 44. 1 20, 3 17. 3 10. 4 7. 9 16-20------------------------------------------ 717 100 52, 8 23. 7 13. 7 5. 3 4. 5 21–25------------------------------------------ 430 100 61. 7 21. 0 9. 1 4. 9 3. 3 26-30------------------------------------------ 260 100 71. 5 19. 3 5. 8 1. 9 1. 5 31-35------------------------------------------ 160 100 74. 4 13. 1 8. 1 2. 5 1. 9 36-40------------------------------------------ 63 100 74. 6 12.7 6. 3 3. 2 3. 2 41 and over------------------------------------ 29 100 55. 2 17, 2 13. 8 6. 9 6. 9 1 Covers a shorter period for respondents with fewer than 10 years of ex- perienc €. 2 Data in this table do not correspond to data in table A-7 because of differ- ences in rounding. For this table, fractional years were rounded to the next which were rounded up to 1 year. 3 Excludes 106 respondents who did not report years of experience. 37 TABLE A–9.—Distribution of persons in mathematical employment, by age group, for each type of organizational unit, 1960 Percent distribution by age group Number Type of organizational unit reporting - Total | Under 25–29 30–34 35–39 40–44 45–49 50–54 55–59 60–64 || 65 years 25 years | years years years years years years years years and Over All types of units— — — — — 1 9, 669 100 9. 1 || 26. 0 || 25. 2 | 17. 3 9. 5 5. 9 || 4. 0 1. 9 0. 8 0. 3 Computing laboratory------- 2, 232 100 | 13. 7 || 36. 0 || 26. 2 | 13. 7 || 5. 7 2. 6 1. 6 4 |------ ... 1 Mathematics unit----------- 1, 197 100 | 1.1. 1 || 32. 4 || 23. 2 | 17. 0 || 7. 6 3. 5 2. 8 1. 3 ... 8 ... 3 Statistical unit-------------- 751 100 5. 6 | 19. 2 || 23. 2 | 18. 2 | 13. 4 9. 6 6. 5 2. 7 . 9 ... 7 Engineering unit------------ 1,471 100 | 8. 4 || 26. 2 | 28.9 || 17. 9 7. 7 || 4, 6 || 3. 5 | 1.9 ... 6 ... 3 Operations research unit ----- 774 100 6. 6 22. 5 28. 3 || 20. 4 || 12. 8 5. 0 2. 5 . 9 ... 4 ... 6 Research laboratory--------- 853 100 7. 3 22. 7 || 24, 1 || 19. 3 || 11. 7 7, 2 4. 2 1. 6 1. 4 . 5 General technical staff------- 931 100 5. 4 || 16. 5 || 25. 6 17. 9 || 12. 8 9. 5 6. 3 4. 1 1. 3 ... 6 Production unit------------- 163 100 9. 8 || 25. 8 || 22. 7 | 18. 4 9. 2 8. 0 1. 8 4. 3 |------|------ Administration unit - - - - - - - - - 353 100 1. 1 6. 2 15. 9 22. 2 | 19. 8 || 17. 0 || 11. 0 5. 4 1. 1 ... 3 ther---------------------- 944 100 9. W .22. 7 || 23. 5 || 17. 7 9. 0 7. 5 6. 0 2. 1 1. 6 ... 2 ! Excludes 313 respondents who did not specify age or organizational unit. TABLE A-10.-Educational level by organizational unit for persons in mathematical employment, 1960 Percent distribution by educational level Number Type of organizational unit reporting Total Doctor’s Master’s Bachelor’s No degree degree degree degree All units------------------------------ 1 9, 765 100 7. 2 25. 7 61. 3 5. 8 Computing laboratory------------------------ 2, 265 100 3. 0 22. 6 70. 2 4, 2 Mathematics unit---------------------------- 1, 219 100 13. 3 26. 7 57. 0 3. 0 Statistical unit------------------------------ 754 100 9. 0 29. 0 53. 5 8. 5 Engineering unit----------------------------- 1, 498 100 3. 5 25. 8 64. 6 6. 1 Operations research unit---------------------- 782 100 12. 1 35. 3 50. 3 2. 3 Research laboratory-------------------------- 868 100 18. 3 29, 3 49. 1 3. 3 General technical staff------------------------ 938 100 6. 0 24, 8 61. 4 7. 8 Production unit----------------------------- 166 100 ... 6 12. 0 78. 4 9. O Administration unit-------------------------- 337 100 2. 7 23. 7 58. 2 15.4 ther-------------------------------------- 938 100 3. 9 21. 7 65. 3 9. 1 1 Excludes 217 respondents who did not specify educational level or type of organizational unit. 38 TABLE A-11.-Operational status of persons in mathematical employment, by age and educational level, 1960 - Percent distribution by operational status Age group 1 ; . - reporting inis- - h - - Total | * | supervisor |*::::::::::::::" All educational levels Total------------------------------------------- 28, 817 100 8. 3 20.4 71. 3 Under 25 years----------------------------------------- 801 100 ... 6 2. 0 97. 4 25–29------------------------------------------------- 2, 292 -100 2. 0 10. 2 87. 8 30-34------------------------------------------------- 2, 222 100 5. 4 23. 0 71.6 36-39------------------------------------------------- 1, 522 100 10. 2 31. 8 58. 0 40-44------------------------------------------------- 840 100 16. 8 28. 4 54, 8 45-49------------------------------------------------- 525 100 23. 6 30. 1 46. 3 50-54------------------------------------------------- 355 100 25. 4 30. 7 43. 9 55-59------------------------------------------------- 171 100 26. 3 21. 1 5.2. 6 60-64------------------------------------------------- 61 100 21. 3 19. 7 59. 0 65 and over------------------------------------------- 28 100 0 25. 0 75. 0 Doctor's degree Total------------------------------------------- 659 100 9. 26.4 64. 0 Under 25 years-----------------------------------------|----------|----------|----------|----------|---------- 25-29------------------------------------------------- 59- 100 1. 7 15. 3 83. 0 30-34------------------------------------------------- 171 100 4. 1 20. 5 75. 4 36-39------------------------------------------------- 197 100 9. 1 33. 0 57. 9 40-44------------------------------------------------- 89 100 13. 5 34.8 51. 7 45-49------------------------------------------------- 67 100 17. 9 26. 9 55. 2 50-54------------------------------------------------- 43 100 20. 9 34.9 44, 2 55-59------------------------------------------------- 19 ----------|----------|-------------------- 60-64------------------------------------------------- 11 ----------|----------|----------|---------- 65 and over-------------------------------------------- 3 ------------------------------|---------- Master’s degree Total------------------------------------------- 2, 232 100 8. 0 24, 6 67. 4 Under 25 years----------------------------------------- 56 100 ----------|---------- 100. 0 25–29------------------------------------------------- 462 100 ... 6 11. 9 87. 5 30-34------------------------------------------------- 622 100 5. 6 24. 9 69. 5 36-39------------------------------------------------- 514 100 9. 9 34. 0 56. 1 40-44------------------------------------------------- 281 100 13. 5 29. 9 56. 6 45-49------------------------------------------------- 142 100 19. 7 33. 8 46. 5 50-54------------------------------------------------- 94. 100 19. 1 30.9 | 50. 0 55-59------------------------------------------------- 42 100 - 11. 9 19. 0 69. 1 60-64------------------------------------------------- 12 ----------|----------|-------------------- Bachelor’s degree Total------------------------------------------- 5, 304 100 7. 1 17. 5 75. 4 Under 25 years----------------------------------------- 705 100 ... 7 2. 0 97. 3 25–29------------------------------------------------- 1,672 100 2. 3 9. W 88. 0 30-34------------------------------------------------- 1, 315 100 4. 8 22. 8 72. 4 35-39------------------------------------------------- 697 100 10. 0 31. 0 59. 0 40-44------------------------------------------------- 391 100 15. 3 26. 9 57. 8 45-49------------------------------------------------- 234 100 24. 4 26. 5 49. 1 50-54------------------------------------------------- 167 100 27. 6 28. 1 44. 3 55-59------------------------------------------------- 83 100 37. 3 22. 9 39. 8 60-64------------------------------------------------- 28 100 35. 7 17. 9 46.4 65 and over-------------------------------------------- 12 --------------------|----------|---------- No degree Total------------------------------------------- 488 100 17. 4 22. 6 60. 0 Under 25 years----------------------------------------- 37 100 ---------- 2. 7 97. 3 5–29------------------------------------------------- 85 100 2.4 8, 2 89.4 30-34------------------------------------------------- 93 100 15. 1 21. 5 63. 4 35-39------------------------------------------------- 91 100 14. 3 25. 3 60. 4 40-44------------------------------------------------- 59 100 32. 2 27. 1 40. 7 45-49------------------------------------------------- 62 100 33. 9 37. 1 29. 0 50-54------------------------------------------------- 29 100 37. 9 34. 5 27. 6 55-59------------------------------------------------- 19 |----------|--------------------|---------- 60-64------------------------------------------------- 10 ----------|----------|----------|---------- 65 and over-------------------------------------------- 3 |----------|----------|----------|---------- 1 Percents not computed for age groups containing fewer than 20 respondents. 2 Excludes 1,165 respondents—868 who specified “other” for operational status and 297 others who did not specify operational status, age, or educa: tional level. Total is larger than the detail by educational level because it includes respondents who did not Specify educational level. 39 TABLE A-12.-Operational status by type of organizational unit of persons in mathematical employment, 1960 Percent distribution Mathematical practitioners who work: other : Number opera- Type of organizational unit reporting - * ... • - tional - Total | Adminis- || Super- | Primarily Individually As an As a member | Status trators visors | with other | with little individual of a team made mathema- Or 110 COIl- consultant up mostly of ticians Sultation to others nonmathe- with others maticians i All types of units----------- 1 9, 870 100 7.6 | 18.5 23.7 12, 9 || 10. 5 18. 1 | 8. 7 Computing laboratory------------ 2, 278 100 4. 2 | 15. 5 || 37.8 16. 6 8. 3 10. 7 || 6.9 athematics unit---------------- 1, 227 100 1. 6 || 11. 3 44. 6 21. 0 11. 7 6. 8 3. 0 Statistical unit------------------- 770 100 6. 4 || 22. 7 18. 6 9. W 22, 6 12. 5 7. 5 Engineering unit----------------- 1, 504 100 3. 3 || 20. 9 9. 4 9. 8 7. 1 37. 0 | 12. 5 Operations research unit ---------- 787 100 4. 6 19. 3 21. 2 10. 9 13. 4 22. 5 8. 1 Research laboratory-------------- 870 100 4. 1 || 21. 0 16. 4 17. 5 8, 9 22. 7 9. 4 General technical staff------------ 947 100 4. 1 || 21. 0 16.4 17. 5 8. 9 22. 7 9.4 Production unit------------------ 165 100 14. 6 || 30. 3 21. 8 7. 3 4. 2 17. 0 4, 8 Administration unit-------------- 357 100 4.7. 9 || 17. 6 ... 6 4, 8 7. 8 11. 5 9, 8 Other--------------------------- 965 100 14, 2 | 19. 1 17. 4 7.7 7. 4 18. 0 16. 2 1 Excludes 112 persons who did not specify organizational unit or operational status. TABLE A-13–Mathematics education cited as minimum requirement for current position, by educational level and major subject field, for persons in mathematical employment, 1960 Mathe- |Other physical Mathe- |Other physical Minimum mathematics education required Total matics science or Other Total matics science or Other engineering engineering Doctor’s degree Master’s degree Number reporting-------------------- 708 438 211 59 || 2, 503 | 1,568 610 || 325 Percent distribution: Total.------------------------- 100. 0 100. 0 100. 0 100. 0 100. 0 100. 0 100. 0 || 100. 0 Doctor's degree------------------ 46. 2 62. 8 20. 4 15. 2 3. 4 4. 1 2. 8 . 9 Master's degree------------------ 36. 2 29. W 52. 1 27. 1 34. 4 40. 6 31. 0 10. 5 Bachelor’s degree----------------- 12. 0 6. 4 18. 0 32. 3 45. 5 46. 2 41. 9 49. 8 Less than a college major in mathe- matics------------------------ 5. 6 1. 1 9. 5 25. 4 16. 7 9. 1 24, 3 38. 8 Bachelor’s degree No degree 1 Number reporting-------------------- 5, 936 || 4, 232 1, 224 480 545 241 142 162 Percent distribution: Total.------------------------- 100. 0 100. 0 100. 0 100. 0 100. 0 100. 0 100. 0 || 100. 0 Doctor's degree------------------ ... 8 ... 7 ... 7 1. 0 1. 1 2. 5 ---------------- Master's degree------------------ 10. 1 9. 3 14. 2 6. 7 6. 2 6. 2 8. 5 4. 3 Bachelor's degree----------------- 60. 9 67. 7 47. 1 36. 9 43. 3 57. 3 34. 5 30, 2 Less than a college major in mathe- matics------------------------ 28. 2 22. 3 38. 0 55. 4 49. 4 34. 0 57. 0 65. 5 1 Major subject field classified according to field of greatest number of credits. 40 TABLE A–14—Major mathematical function of persons in mathematical employment, by employer, 1960 Percent distribution by function Applied re-| Nontech- Technical Number Basic re- || Search and nological | Tech- Services | Teach- Employer report- search in develop- research, nical allied to ing e ling Total the natural ment in including | Services | Sales, pro- and | Admin- || Other sciences the natural marketing allied Imotion, train- |istration and en- Sciences and other to pro- or dis- ing gineering and en- economic | duction | tribution gineering research All employers-------------- 19, 867 100 7. 0 45.4 4. 0 | 18, 3 4. 4 0.8 8, 2 11. 9 Private industry------------------ 7, 103 100 4, 8 45. 6 3. 9 21. 2 5. 8 ... 8 7. 3 10. 6 Aircraft and parts--------------- 1, 968 100 7. 4 59. 6 1. 0 || 19, 9 1. 4 1. 1 3. 2 6.4 Transportation equipment, (ex- \ cept aircraft)----------------- 212 100 5. 2 66. 0 1. 9 || 13. 7 1. 4 ... 5 5. 2 6. 1 Electrical equipment------------ 1, 224 || 100 4. 7 58. 2 2. 4 || 18. 0 2. 8 || 1. 1 || 5, 1 7. 7 Machinery (except electrical)----- 594 100 5. 2 46. 8 2. 9 || 20. 0 9. 3 1. 0 3. 9 10. 9 Professional and scientific in- struments-------------------- 188 100 6. 9 56, 9 1. 6 || 18. 1 5. 9 |------ 3. 2 7.4 Other durable manufacturing----- 525 || 100 2. 3 46.8 2. 3 || 25, 9 10. 1 . 8 || 3.4 8.4 Petroleum products and extrac- -- tion------------------------- 452 100 6. 4 39. 4 7. 5 || 32. 5 2. 6 ------ 5. 8 5. 8 Chemicals and allied products---- 313 100 8. 3 42. 5 7. 7 21. 7 5. 1 1. 6 5. 4 7.7 Other nondurable manufacturing-- 168 100 2.4 32. 1 14. 9 || 25. 1 8. 9 ... 6 8. 9 7. 1 Insurance---------------------- 914 100 . 1 2. 9 5. 0 | 19. 6 17. 4 . 3 || 25. 5 29. 2 Other non-manufacturing-------- 545 100 2. 4 34. 7 11. 9 || 25. 3 4. 6 ... 2 8. 1 12. 8 Federal Government-------------- 2, 542 100 11. 4 44, 8 3. 6 || 11. 7 ... 8 . 8 || 11. 1 15. 8 Army------------------------- 971 100 5. 7 40. 2 2. 6 || 12. 2 1. 5 1. 0 || 21. 9 14. 9 Wavy-------------------------- 577 100 9. 4 59. 1 1. 7 | 13. 7 -------- ... 3 3. 3 12. 5 Air Force---------------------- 454 100 7. 9 49. 8 2. 4 || 14. 1 ... 8 ... 2 4, 8 20. 0 National Aeronautics and Space Agency---------------------- 229 100 44. 5 46. 3 |-------- 8, 1 -------- . 4 1. 3 4. 4 Commerce--------------------- 105 100 18. 1 19. 0 13. 3 | 12. 4 |-------- 1. 0 3. 8 32.4 Other agencies------------------ 206 100 11. 2 27.2 15. 0 || 7. 8 . 9 2. 4 || 10. 7 || 24.8 Nonprofit organizations------------| 222 || 100 25. 7 50. 4 10. 4 || 2.2 ... 5 . 9 || 2. 2 7.7 1 Excludes 115 respondents who did not specify their major function. 41 TABLE A-15.-Median and quartile annual incomes of ‘. and women in mathematical employment, by age and educational evel, 1960 1 Total Men Women Age group Number | First Median || Third | Number | First Median | Third | Number | First || Median | Third reporting quartile quartile | reporting | quartile Quartile | reporting quartile quartile ` All educational levels All age { - groups?--|. 9, 442 $6,900 $8,500|510, 900 7, 996 $7, 300 $8,900|$11,400 1, 366 $5,800 $6,600 $7,800 Under 25 years--- 851] 5, 400 6, 100 6, 507 5, 700 6, 400 6, 900 341| 5, 200 5, 700 6, 300 25–29------------ 2, 438 6, 400 7, 300 8, 500 2,041 6, 500 7,400 8, 600 389 6,000 6, 600 7, 500 30-34------------ 2, 299| 7, 600. 9, 100 10, 900] 2, 118 7, 800 9, 200 11, 000 179| 6, 300 7, 100 8, 500 35-39------------ 1, 571 8, 200 10, 300 12, 500 1,416 8,600 10, 500 12, 800 150 6, 300 7, 300 8, 600 40–44------------ 883 8, 400 10, 500 13, 600 766 8, 900 10, 900 14, 100 112 6, 400 7, 600. 9, 200 45–49–----------- 548 8, 600 10, 900 14, 200 475|| 9, 200 11,400 14, 800 72 6, 800 7, 800 9, 000 50–54------------ 372 8, 400 11, 000 15, 100 311| 9, 000 12, 200 17, 000 59 6, 400 7, 600. 9, 200 55-59------------ 174, 8,600 11, 600 16, 400 149 9, 100 12, 300 17, 500 25 6, 500 8, 400 10, 300 60–64------------ 69| 8, 500 11, 200 14, 000 60 9,000 11, 400 14, 500 9|--------------------- - Doctor’s degree All age - grOupS.--- 656|$11, 200813, 000$15,000 627|$11,300|$13, 100815, 200 24|$10, 000$11, 000$12,500 Under 25 years---|-------|-------|-------|-------|--|-- ---------------------------------------------|------- 25–29------------ 64 9, 700 10, 800 11, 900 63| 9, 700 10, 800 11, 900 !--------------------- 30-34------------ 168] 10, 800 12, 100 14, 000 163| 10, 800 12, 200 14, 000 5'-------|-------------- 35-39------------ 188 12, 100 13, 700 15, 900 184 12, 200 13, 800 16, 000 4--------------------- 40–44------------ 88 11, 900 14, 300|| 17, 500 79| 12, 200 14, 600 17, 700 6--------------------- 45-49------------ 66 11, 800 14, 200 18, 200 63| 11, 900 14,400 18, 400 3--------------------- #3; - - - - - tº- ºr - " - - - - # 11, 100 13, 800 18, 400 ; 11, 600 14, 100 18, 900 3|--------------------- T*) tº — — — — — — — — — — — — 6 !--------------|------- 60–64------------ 9 }11, 600 13, 100 14, 900 8 }11, 700 13, 200 15,000 !-------|-------------- Master’s degree All age | groups---| 2, 362 $8, 200 $9,900; $11,900] 2, 146|| $8,400|$10, 100 $12, 200 197 $6,900 $8,000 $9,300 Under 25 years--- 63 6, 300 7, 300 7, 900 52 6, 700 7, 400 8, 000 11|--------------------- 25–29------------ 491 7, 400 8, 300. 9, 400 437| 7,400 8, 300. 9, 500 52 6, 800 7, 600 8, 600 30-34------------ 632, 8, 500. 9, 900 11, 500 596, 8, 700 10, 000 11, 600 35 6, 800 7, 900. 9, 400 35–39–----------- 526 9, 300 11, 100 13, 100 494. 9, 500 11, 200 13, 200 30| 7, 100 7, 800 9, 600 40–44------------ 284 9, 400 11, 400 14, 100 263| 9, 700 11, 700 14, 300 *} 8, 100 8, 900 10, 100 45–49–----------- 148 9, 600 11, 500 14, 700 133 10, 100 11, 800 15, 200 15 y j 3. 50–54------------ 99| 8, 500 11, 200 14, 500 82 8, 900 12, 300 15, 300 *} 6, 900 8, 700 10, 000 55-59------------ 46 8, 500 10, 500 13, 500 37| 9, 100 11, 300|| 14, 500 9 y y 3. 60–64------------ 14--------------|------- 12|--------------|------- 2--------------|------- Bachelor’s degree All age groups---| 5, 717| $6,500 $7, 700 $9,700. 4, 609 $6,700 $8, 100|$10, 200 1,065 $5,700 $6,500 $7,400 Under 25 years--- 750, 5, 400 6, 100 6, 700 426 5, 700 6, 300 6,800 320 5, 200 5, 700 6, 300 25–29------------ 1, 778 6, 200 7,000 8, 000 1, 452 6, 300 7, 100 8, 200 322 5, 900 6, 500 7, 300 30–34–----------- 1, 374 7, 200 8, 500 10, 100 1, 243| 7,400 8, 700 10, 300 130 6, 200 6, 900 8, 000 35–39–----------- 732 7, 700 9, 300 10, 900 631| 8, 100. 9, 700 11, 200 99| 6, 300 7, 100 8, 300 40–44------------ 419 7, 700 9, 700 11, 900 342 8, 500 10, 200 12, 800 76 6, 200 7, 300 8, 800 45–49–----------- 239| 8, 300 10, 200 13, 500 195| 9,000 11, 000 14, 400 44 6, 500 7, 500 8, 600 50–54–----------- 175 8, 000 10, 800 15, 800 135|| 9, 600 12, 500 18, 800 39| 6, 200| 7, 100 8, 600 55-59------------ 83| 8, 500 13, 100 18, 700 72 9, 100 14, 200 19, 400 11|--------------------- 60–64------------ 33| 9, 100 11, 600 15, 900 28, 10, 000 12, 500 17, 300 5'-------|-------------- No degree All age - grOupS.--- 549 $6,800 $7,900 $9,500 479 $7, 100 $8,000 $9,900 63 $5,600 $6,400 $7, 200 Under 25 years--- 35 4, 700 5, 500 6, 400 27| 4, 800 5, 700 6, 500 8--------------------- 25–29–----------- 89 6,000 6, 700 7, 600 77| 6, 100 6, 700 7, 700 11|--------------------- 30-34------------ 101| 7, 000 7, 700 8, 600 92| 7, 100 7, 700 8, 700 9|--------------------- 35–39–----------- 102 7, 200 7, 900 8, 800 88 7,400 8, 100 8, 900 13|-------|-------------- 40–44------------ 70 7, 900 8, 900 11, 100 63| 8, 100. 9, 100 11, 600 7|-------|-------------- 45–49–----------- 73| 7, 600 8, 900 11, 000 65 7, 700 9, 300 11, 200 7|--------------------- 50–54------------ 32 8, 400 10, 000 14,000 # 8, 500 10, 100 14, 300 # * - - - - - - - sº * * - - - - - as ess sº - - - - 55-59------------ 19 16|l o anal a coal onalſ 3-------|-------|------- 60–64------------ {} 7,700 9, 100 11, 600'ſ #} 8,000 9,600 11, 800% 1|--------------------- 1 Quartile and median incomes not computed for cells with fewer than 20 respondents. 2. The total for all age groups includes respondents who did not Specify age and also those 65 or over for which no data are shown because fewer than 20 provided information on income. 42 TABLE A-16.-Median and quartile annual incomes 1 of persons in mathematical employment, by principal type of em- ployer, educational level, and age, 1960 Private industry excluding insurance Insurance industry Federal Government A ge Numberſ First Median Third || Number| First Median Third Numberl First Median Third reporting quartile quartile ||reporting quartile quartile reporting quartile quartile Doctor’s degree Under 25 years--|------|---------------------|------------------------------|--------------------------- 25–29–---------- 55 $9,800|$10, 9001311, 800||--|--|--|--|--|--|---------|--------- 5|-------|-------|------- 30-34----------- 121 11, 300 12, 600 14, 200||------|--|--|--|---------|--------- 35| $9,400|$10,600|$11,700 #3; * * = m = <= me ses, sº arm as 143| 12, 500 14, 100 16, 800 !------------------------ 32 11, 100 12,000. 13, 500 Tºtt--- - - - - - - - - 66 11, 900 14, 800 17, 800 2------|---------|--------- 12 * ; :----------- tºº tºº is sº !------------------------ § 11, . º ſº º º Tººt--— — — — — — — — — 18|l on = onol a cool------------|------------------ 2 11, 1 y y 55-59----------- #}11, 100 15, 300 19, 800 *------------------------ 4-------|-------|------- 60–64----------- 4--------------|-------|------|------|---------|--------- 5|-------|-------|------- Master’s degree Under 25 years-- 50 $7,000 $7,500 $8, 200 2------------------------ 7|-------|-------|------- 25–29–---------- 385 7, 600 8, 500 9, 600 19|------------------------ , 66 $6,300 $7, 100 $8,100 30-34----------- 445, 8,800 10, 000 11, 600 46|$8,800|$10, 300 |$13, 200 114 7, 600 8, 900 10, 900 35–39----------- 348 9, 600 11, 300 13, 400 45|10, 300 13, 100 || 17, 000 122 8, 300 10, 100 11, 400 40–44----------- 158 10, 000 12,000 14, 700 23|11, 900 15, 500 220,000+ 97 8, 500 10, 100 11, 900 45–49–---------- 77| 10, 200 12,000 14, 700 2015, 500 18, 000 220,000+ 49, 8,700 10, 500 11, 700 50–54----------- 29, 8,900 12, 500 14, 900 19|------|---------|--------- 51| 7, 800 9, 600 11,600 55-59----------- 19|-------|-------|------- 8------------------------ 19|-------|-------|------- 60–64----------- 5'--------------------- 2------------------------ 7|-------|-------------- Bachelor’s degree Under 25 years--| 513 $5,600 $6,300 $6,800 64|$5, 200 $5,900 $6,700 156 $5, 200 $5,600 $6,000 25–29----------- 1, 134 6,400 7, 200 8, 200 169| 6,000 7, 100 8, 600 449 5, 700 6, 500 7, 300 30-34---------- - 869 7, 300 8, 700 10, 100|| 141 8, 200 10,000 | 12, 700 337 6, 800 7, 900. 9, 000 35-39----------- 447| 7, 900. 9, 600 11, 000 65 9, 400 12, 300 | 15, 300 203| 7, 100 8,400. 9,900 40–44----------- 199| 8, 200. 9, 900 11, 600 58|12, 600 16, 600 220,000+ 160 7, 100 8, 400 10, 200 45–49–---------- 97 8, 400 10, 100 13, 100 44|13, 500 18, 500 220,000+ 98 7, 600 8, 900 10, 800 50–54–---------- 55 8, 800 10, 900 13, 900 47|15, 400? 20,000+|220,000+ 72 6, 800 8, 100 10, 300 55-59----------- 26, 8,600 12,000 17, 100 36|14, 100 17, 900 220,000+ 22 6, 700 7, 700 9,600 60–64----------- 13|--------------|------- 9|---------------|--------- 11|-------|-------|------- No degree Under 25 years – 24 $5,000 $6,000 $6,700 3------------------------ 8-------|-------------- T4 º’- - - - - - - - - - - 67| 6, 100 6, 700 7, 800 2-------------|----------- 19|----------- - - - 1 - - - - - - - 30-34----------- 65 6, 900 7, 700 8, 700 5'------------------------ 31|| $7, 100 $7,600 $8, 200 35–39----------- 49, 7, 100 7, 800 9, 300 6------------------------ 47| 7, 300 8, 000 8, 600 40–44----------- 23| 8, 900 10, 400 12, 900 8------------------------ 38 7, 300 8, 300 9,000 45–49–---------- 17|-------|-------------- 6-------------|----------- # 7,400 8, 200. 9, 700 50–54----------- 7|--------------------- 7|------------------------ 18| , 55-59----------- 1|--------------------- 7|------------------------ #} 7, 900 8, 600|| 9,900 60–64----------- 4-------|-------|------- 2------------------------ 6-------|-------|------- 1 Quartile and median incomes were not computed for cells with fewer than 20 replies. 3 The median and third quartile incomes shown as $20,000 fell within the open-end income class of $20,000 and over. 43 * * TABLE A-17.-Median annual incomes of persons in mathematical employment, by age, educational level, and type of organ- ~. izational unit, 1960 1 Computing laboratory Engineering unit Mathematics unit General technical staff Administration unit Age group - Master’s | Bachelor’s | Master’s | Bachelor’s | Doctor’s | Master’s | Bachelor’s | Master’s | Bachelor’s | Master’s | Bachelor’s degree degree degree degree degree degree degree degree degree degree degree Under 25 years------|-------- $6,100 |--------| $6,300 |--------|-------- $6,000 |_ _ _ _ _ _ _ _ $5, 700 |--------|-------- 25–29-------- $8, 200 6, 900 $8,700 7, 200 |-------- $8,000 | 6, 800 $8,400 , 600 --------|-------- 30–34–------- 9, 800 8, 200 10, 100 8, 600 |$11,900 9, 500 8, 100 10, 200 9, 600 |_ _ _ _ _ _ _ _ $8,900 35–39-------- 11, 000 9, 100 | 11, 800 9, 300 || 13,400 || 10, 400 8, 800 || 11, 600 | 10, 600 || $9,900 10, 600 40–44-------- 10, 500 8, 600 || 12, 800 9, 400 }1 4, 200 || 9, 800 || 9, §99 || 14, 199 || || 899 |\11, 800 || 9, 299 45–49–------- 10, 500 || 8, 600 }11 700 { 9, 600 y }11 000 | 8, 200 || 14,000 | 12,800 y 10, 600 50-54--------|--------|-------- y 10, 200 |-------- ' ' -------- 14, 100 |{{# 299 ||-------- 12,000 *~39---------------------------------------------------------------- y 16, 700 --------|-------- Research laboratory Operations research unit Statistical unit Produc- Other units tion unit Doctor’s | Master’s | Bachelor’s | Doctor’s | Master’s | Bachelor’s | Master’s | Bachelor’s | Bachelor’s | Master’s | Bachelor’s degree degree degree degree degree degree degree degree degree degree degree Under 25 a Years------|--------|-------- $5,900 --------|-------- $5,800 |_ _ _ _ _ _ _ _ $5,500 --------|-------- $6,000 25–29__ _ _ _ _ _ _ $10, 300 || $8,400 7, 100 |-------- $8,300 7, 200 $7, 700 , 500 $6,300 $8, 200 6,900 30–34–------- 12, 500 9, 600 || 8, 800 |$11,900 | 10, 200 9, 000 | 10, 000 8, 100 8, 100 | 10, 100 8, 600 35–39__ _ _ _ _ _ _ 14, 300 11, 400 8, 500 | 13, 500 | 11, 700 | 11, 000 9, 400 8,500 9, 000 || 10, 800 8, 700 40–44__ _ _ _ _ _ _ }14 300 12, 800 8, 500 |-------- 12,400 | 11, 900 || 10, 600 9, 300 |-------- 11, 100 10, 100 45–49–------- y 12, 900 || 9, 500 --------|--------|-------- 11, 000 || 8, 500 -------- , - 14, 200 ; : * = - sºme " me mºs. e-m - * * = * * * * ' ' ------------------------|-------- ' ' || ---------------|-------- 17, 500 1 Median incomes not computed for cells with fewer than 20 respondents. TABLE A–18.-Distribution by income bracket and principal type of employer of persons in mathematical employment, 1960 Private industry Annual income . - Federal Nonprofit All private |Private indus- Government | Organizations industry try excluding | Insurance IIlSUIran Ce Total------------------------------------------- 100. 0 100. 0 100. 0 100. 0 100. 0 Under $4,000------------------------------------------ . 1 | ... 1 ... 2 ... 2 . 5 $4,000-$4,999------------------------------------------ 1. 4 1. 1 2. 8 3. 0 5. 8 $5,000-$5,999----------------------------------------- 6. 8 6. / 7. 6 12. 7 10. 6 $6,000-$6,999–----------------------------------------- 15. 5 16. 5 9. 2 17. 5 13. 0 $7,000-$7,999------------------------------------------ 15. 8 16, 9 9. 2 18. 6 11. 5 $8,000-$8,999------------------------------------------ 12. 6 13. 3 8. 1 16. 7 4. 3 $9,000-$9,999------------------------------------------ 10. 9 11. 4 7. 3 7. 8 11. 1 $10,000-$10,999–--------------------------------------- 9. 5 10. 0 6. 2 8. 0 10. 6 $11,000-$11,999–--------------------------------------- 6. 8 7. 1 5. 2 8. 0 7, 2 $12,000-$14,999–--------------------------------------- 11. 6 11. 1 14, 8 7. 0 13. 9 $15,000-$19,999–--------------------------------------- 6. 0 4. 7 14. 4 . 5 7. 2 $20,000 and over--------------------------------------- 3. 0 1. 1 15.0 ---------- 4, 3 44 & TABLE A–19.-Median annual incomes of supervisors and nonsupervisors in mathematical employment in private industry and in Government, by age and educational level, 1960 1 - Private industry, excluding - InSUiT3T1C0. Insurance Federal Government Age group Supervisor Nonsupervisor Supervisor Nonsupervisor Supervisor Nonsupervisor All educational levels All age groups------------------- $10,800 $7,800 $13,400 $7, 700 $9,400 $7, 100 Doctor’s degree All age groups------------------- $14,800 $12, 200 ----------|------------ $12, 200 $11,500 Under 25 years------------------------|----------|------------|----------|------------|----------|------------ T**7 — — — — — — — — — — — — — — — — — — — — — — — — — — — - - - - - - - - - - - - - - - 10, 900 ----------|------------|----------|------------ 30-34-------------------------------- 13, 500 12, 100 ----------|------------|---------- 10, 000 35-39-------------------------------- 15, 200 13, 400 ----------|------------|----------|------------ 40-44-------------------------------- 15, 600 13, 500 ----------|------------|----------|------------ 45-49-------------------------------- 17, 700 ------------|----------|------------|----------|------------ Master's degree All age groups------------------- $11,500 $9,100 $14,900 $8,900 $10,600 $8,300 Under 25 years------------------------|---------- 7, 500 ----------|------------|----------|------------ 25-29-------------------------------- 9, 500 8, 200 ----------|------------|---------- 6,900 30-34-------------------------------- 11, 000 9, 500 11, 700 ------------ 10, 600 8, 100 35-39-------------------------------- 12,000 10, 400 14,000 ------------ 10, 700 9,000 40-44-------------------------------- 13, 200 10, 200 || ſ------------ 10, 900 9, 300 #. ams ºr sm as * = <= amº as sm sºme * * * *ms sº me tº sº sº sº sº as sº sº, º is ºr sºn ºs º ºs 13, 200 10, 700 |} 19, 400 ||------------ # : ; #: 50– * sm am, sº sº, º sº. º. ººm s emº, º smºs me sºme sº * * * * * * * * * * * * * * * *= mºs = * * * * * * * * : * * º . . . . . . . * = s gºs ºs sº sº tº ºms as sºns s y y 55-59--------------------------------|---------- } 10,300 || ||------------|..."4" | ** Bachelor’s degree All age groups------------------- $9,800 $7, 200 $12,700 $7,300 $8,900 $6,700 Under 25 years------------------------|---------- 6, 300 |---------- 5, 700 ---------- 5, 600 25–29-------------------------------- 8, 300 7,000 7, 700 6, 700 7, 700 6, 300 30-34-------------------------------- 9, 800 8, 200 10, 600 9, 100 8,800 7, 400 35-39-------------------------------- 10, 400 8, 800 12, 500 ------------ 9, 400 7, 700 40-44-------------------------------- 10, 600 9, 100 17, 700 ------------ 9,900 7, 600 45-49-------------------------------- 12,000 8,900 19, 700 ------------ 9,900 8. 400 50-54-------------------------------- 11, 100 10, 000 | * 20, 000+|_ _ _ _ _ _ _ _ _ _ _ _ 9, 200 7, 300 55-59------------------------------------------------------ 19, 100 ------------|----------|------------ 1 Median incomes not computed for cells with fewer than 20 respondents. 2 This median fell within the open end income class of “$20,000 and over.” 45 TABLE A–20–Chief subject matter source of problems, by employer, of persons in mathematical employment, 1960 Percent distribution by chief subject matter source of problems •- - - - - - - : * Biologi- Social Office Employer Number Total Basic |Physics, Earth cal, medi- || Sciences, and | Produc- reporting mathe- chem- || sciences, Engi- cal, and including Insur- businessition and Other matics istry meteor- neering agricul- ſeconomics ance admin- inven- +... • * ology tural and istration] tory Sciences psychology All employers--------- 19,914 | 100 | 18. 3 8. 5 2. 4 | 40. 5 1. 3 2. 1 9. 2 | 5. 1 5. 9 6. Private industry------------ 7, 125 | 100 15.0 7. 6 1. 6 || 46. 0 ... 6 1. 4 | 12. 5 3. 5 || 6. 1 5. Aircraft and parts--------- 1, 971 100 || 17. 1 7. 3 . 7 | 66.4 ... 2 . 2 ------ 1. 5 3. 1 3. Transportation equipment (except aircraft) ---------| 212 100 15. 1 9. 4 |------ 67.5 |_ _ _ _ _ _ _ . 5 ... 5 1. 4 1. 4 4. Electrical equipment------ 1, 228 100 | 19. 1 || 10. 0 .9 53.3 |------- . 9 |------ 3. 3 4. 1 8. Machinery (except elec- $ - - trical) -----------------| 596 100 20. 0 || 5. 9 1. 7 || 47. 1 ... 3 1.5 ------ 4. 0 6.9 || 12. Professional and scientific - - instruments.-------------| 188 100 | 16. 5 || 14. 9 3.2 47.9 |-------|-------|------ 2, 6 || 10. 1 4. Other durable manufactur- • - ink-------------------- 525 100 | 15, 9 6. 0 . 6 || 52.4 ... 4 ... 8 ... 6 4. 0 | 12. 8 6. Petroleum products and extraction-------------- 451 100 17. 1 || 14. 4 || 14 0 || 32.8 |_ _ _ _ _ _ _ 3. 1 1. 1 4. 7 8.4 4. Chemicals and allied prod- - ucts------------------- 315 100 | 11. 1 || 15. 5 1. 3 || 33.3 9, 2 4, 8 ... 6 5. 1 || 13. 7 : 5. Other non-durable manu- - w - - facturing--------------- 169 100 | 16. 6 || 11. 8 |_ _ _ _ _ _ 21, 9 1. 7. 7 |------ 11. 2 21. 3 7. Insurance---------------- 921 100 1. 0 . 1 ... 1 4 |------- . 7 || 94. 2 2. 1 ... 2 1. Mining and construction.-- 26 100 || 11. 6 3. 8 |------ 80.8 -------|------------- 3. 8 ------|------ Other non-manufacturing--| 523 100 || 14. 4 || 4, 2 . 9 || 42.8 . 9 4. 2 1. 5 9. 8 || 13. 5 7. 8 Federal Government-------- 2, 568 100 27, 8 || 10. 0 || 4.5 25.9 3. 2 3. 5 | 1. 1 || 9, 7 || 5. 3 9. 0 Army-------------------- 988 100 26. 9 7. 0 3. 6 22. 1 3. 4 1. 4 |------ 22. 0 5. 8 7. 8 Navy-------------------- 582 100 28, 3 || 17. 4 4. 0 || 34. 5 ... 3 1. 0 |------ 1. 0 4, 6 8. 9 Air Force---------------- 457 100 || 32. 8 4, 8 6. 6 || 27. 6 2. 2 2. 6 . 4 3. 7 8. 1 11. 2 National Aeronatutics and • * - - Space Agency----------| 231 100 | 34.6 | 17.3 2. 2 | 40. 7 |-------|-------|------|--|-- - - I - - - - - - 5. 2 Commerce--------------- 105 100 || 26. 6 || 13. 3 | 16. 2 2.9 |------- 22.9 |_ _ _ _ _ _ . 9 4, 8 || 12. A Other agencies------------ 205 100 | 12. 2 | 6. 3 || 2.0 | 11. 2 | 18. 0 | 16. 1 || 13. 2 || 4.4 || 4, 4 || 12. 2 Nonprofit organizations------ 221 | 100 | 16.3 22.2 2.7 || 30.8 1. 8 7. 7 ------------ 5. 0 || 13. 5 1 Excludes 68 respondents who did not specify subject matter source of problems. TABLE A-21.-Major function, by chief subject matter source of problems, of persons in mathematical employment, 1960 Percent distribution by major function Number Subject matter source of problems reporting Applied Nontech- || Technical | Other Teaching Total Basic research nological sevices technical| and Admin- || Other . researchi and devel- research 2 allied to services 8 training |istration opment 1 production All subject matter sources----- 49, 834 100 || 7.0 45. 4 4. 0 18. 3 || 4, 4 || 0.8 8. 2 11. 8 Basic mathematics----------------- 1, 787 | 100 || 14.5 45. 3 1. 9 19, 0 || 2, 9 | 1. 2 | 3. 3 | 11.9 Physics, chemistry----------------- 843 100 || 21. 5 62. 7 ... 2 6.4 . 5 . 5 || 3. 2 5. 0 Earth Sciences, meteorology--------- 236 || 100 11. 4 61.9 |-------- 15. 2 ------|------ 3. 4 8. 1 Engineering----------------------- 3,997 || 100 || 4, 1 66. 1 ... 7 17, 6 | 1. 7 . 7 || 3. 3 5. 8 Biological, medical and agricultural * sciences------------------------- 129 || 100 || 14. 0 55. 8 3. 1 4. 6 || 3. 1 1. 5 8 17. 1 Social sciences, including economics and psychology------------------ 203 100 1. 5 12. 3 49. 2 3. 9 || 7. 4 || 2. 5 || 3. 0 || 20. 2 Insurance------------------------- 910 || 100 ... 3 2. 5 5. 7 18. 9 17. 0 . 3 || 24, 9 || 30.4 Office and business administration-- 496 100 |_ _ _ _ _ _ 3. 8 12. 3 8. 7 || 8. 1 . 8 54. 2 12. 1 Production and inventory----------- 577 || 100 . 3 6. 2 9. W 59. 1 || 9. 4 . 9 || 5.9 8. 5 9ther---------------------------- 656 100 5. 0 26. 8 7. 9 14, 9 6. 7 . 9 5. 9 31. 9 1 In natural sciences and engineering. * Includes marketing and other economic research. 8 Allied to Sales promotion or distribution. 4 Excludes 148 persons who did not specify major function or chief Subject matter source of problems. 46 TABLE A–22.-Subject matter source of problems cited as first, second, and third most important by persons in mathematica employment, 1960 Most Second Third Most Second Third Subject matter source of problems important In OSt. In 10St. Subject matter Source of problems important most In OSt - - important important important important All respondents.------- 9,982 9, 982 9,982 || Social sciences, including . economics and psychology- 205 205 148. Basic mathematics---------- 1, 815 1, 190 612 || Insurance------------------ 917 41 16. Physics, chemistry---------- 848 1, 434 575 || Office and business admin- Earth sciences, meteorology-- 237 298 221 istration----------------- 502 574 223. Bngineering---------------- 4,015 1, 294 454 || Production and inventory--- 581 577 281 Biological, medical, and agri- Other--------------------- 663 166 82. cultural Sciences---------- 131 46 47 || Not Specified--------------- 68 || 4, 157 7, 323. TABLE A–23.−Mathematical fields used most by persons in mathematical employment, by employer, 1960 Percent distribution by mathematical field used most Number Employer report- Com- |Numericall ing puter analysis, Analytical Probabil- || Actuarial || Logic, Alge- Topology Total tech- theory of mechan- || Analysis ity and mathe- theory bra 2 and Other niques compu- ics 1 statistics matics Of Sets geometry - tation All employers------|*9, 862 100 || 34, 9 9. 5 6. 8 || 16. 8 16.4 8. 7 || 0.8 3. 0 0. 7 2.4 Private industry--------- 7, 089 100 || 36. 1 8. 2 7. 6 15. 1 13. 8 11. 6 1. 0 || 3. 1 . 9 2. 6. Aircraft and parts------ 1, 962 100 || 39, 6 9. 9 13. 5 || 19.8 9. 3 ... 1 . 7 || 3. 0 1. 4 2. 7. Transportation equip- ment (except aircraft)- 241 100 || 38, 9 13. 3 17. 0 || 15. 6 7. 6 ------- 1. 9 || 3. 3 . 5 1. 9, Electrical equipment - - -] 1, 221 100 || 43.9 11. 1 6. 4 || 17. 1 15. 2 ... 2 1. 3 2. 3 ... 3 2. 2: Machinery (except electrical)----------- 596 100 || 50. 7 6. 2 7. 4 || 14, 6 12. 2 ... 2 3. 3 2. 9 . 5 2. 0 Professional and scien- tific instruments.----- 187 100 || 34, 2 16. 0 4, 8 || 18.8 16.0 ------- 1. 6 3. 2 2. 7 2. 7. Other durable manufacturing------- 524 100 || 43. 6 5. 5 9. 0 | 15. 8 18. 1 ... 4 ... 6 3. 6 1. 3 2. l. Petroleum products and extraction------- 446 100 || 47. 8 10. 5 4. 7 || 14, 6 9. 2 1. 3 ... 7 6. 5 ... 6 3. 1. Chemicals and allied products------------ 314 100 29, 9 7. 6 1. 6 11. 5 41. 4 1. 3 ... 3 2. 9 ... 6 2. 9. Other nondurable manufacturing------- 169 100 | 16. 1 4. 7 6. 5 || 21. 9 || 37.8 ... 6 . 6 5. 3 ||------ 6. 5, Insurance------------- 919 100 8. 5 . 5 . 1 1. 5 3. 5 84. 5 ... 1 - 4 ------ . 9. Other nonmanufacturing-- 540 100 || 29, 8 8. 1 5. 0 | 15.9 24. 4 4. 4 ... 6 5. 6 1. 1 5. 0. Federal Government - - - - - 2, 550 100 31, 2 12. 3 4. 4 || 21. 8 || 23.2 1. 5 . 4 || 2.9 ... 3 2. 0, Army----------------- 970 100 || 21. 0 10. 1 3. 8 || 34, 9 25, 2 l------- ... 1 2. 5 ... 4 2. 0. Navy----------------- 582 100 || 38. 3 12. 0 5. 3 | 18. 8 20. 6 ------- ... 7 2. 2 ... 7 1. 4. Air Force------------- 459 100 || 37. 5 14, 6 5. 0 | 16. 1 20. 0 1. 3 1. 2 2, 6 ------ 1. 7 National Aeronautics and Space Agency---- 230 100 || 52. 2 24, 8 6. 1 5. 7 1. 3 . 4 ... 4 6, 5 ------ 2. 6. Commerce------------ 104 100 || 33. 7 7.7 1. 0 3. 8 50.0 -------|------ 3.8 ------------ Other agencies--------- 205 100 20. 5 6. 3 3.4 8. 8 39. 0 15. 1 |------ 2.5 ------ 4. 4. Nonprofit organizations--- 223 100 37. 3 17. 9 6. 7 || 13. 9 || 20.6 ||------- . 4 | 1.8 |------ 1. 4. 1 Includes fluid dynamics and electromagnetics. 3 Excludes 120 respondents who did not specify mathematical field used * Includes theory of numbers. most. 47° TABLE A–24.—Mathematical fields used most, by chief subject matter source of problems, of persons in mathematical employ- ment, 1960 Percent distribution by mathematical field used most Subject matter source of problems Number Com- || Numeri- - Topol- reporting puter |cal analy-| Analyt- || Analy- | Probabil- || Actuarial | Logic Alge- |ogy and Total tech- |sis theory ical me- sis ity and mathe- theory bra 2 geom- || Other niques of com- chanics 1 statistics matics of sets etry putation All subject matter SOUII'CeS - - - - - - - - - 39, 835 100 || 34.9 9. 5 6. 8 || 16. 8 16.4 8, 7 0, 8 3. 0 0. 7 2.4 Basic mathematics------- 1, 804 100 || 46. 0 14, 7 1. 9 || 17. 5 10. 8 7 1. 2 4. 0 ... 8 2.4 Physics, chemistry - ------ 840 100 32. 7 17. 5 14. 3 | 18. 2 12. 1 1 ... 1 2. 6 ... 7 1. 7 Earth sciences, mete- orology--------------- 234 100 || 46. 2 12.4 6. 4 || 16. 7 4. 3 9 . 4 5. 1 3. 8 3. 8 Engineering------------- 3,985 100 || 37.8 || 10.6 | 12. 2 | 19. 0 || 13.2 1 . 8 || 3. 2 | . 9 2. 2 Biological, medical and agricultural sciences---- 131 100 | 6.9 3.0 ------- 18. 3 69. 4 |------- ... 8 • 8 ------ ... 8 Social sciences, including economics and psy- chology--------------- 202 100 || 14. 4 1. 0 |------- 13. 3 62. 9 4. 9 . 5 2.5 |------ . 5 Insurance--------------- 915 100 7. 2 . 3 |-------| 1. 1 3. 6 || 86. 7 . 1 4 |------ ... 6 Office and business ad- ministration----------- 489 100 || 30, 2 2.9 |------- 31. 5 24. 7 2. 5 ... 2 3. 1 |------ 4, 9 Production and inventory- 578 100 || 36. 0 2. 6 . 3 || 15. 6 || 37. 7 . 3 1. 2 || 4, 3 ... 3 1. 7 Other------------------- 657 100 || 38. 1 5. 2 1. 7 || 14. 1 28. 2 2. 7 1. 5 1. 4 ... 6 6, 5 1 Includes fluid dynamics and electromagnetics. 3 Excludes 147 respondents who did not specify the mathematical field * Includes theory of numbers. used most Or Subject matter Source of problems. * TABLE A-25–Mathematics courses cited by respond ents as minimum requirements for their positions, by chief subject matter source of problems, 1960 Subject matter Source of problems Biologi- Social Courses cited Earth cal, Sciences Office | Produc- Basic | Physics, Sciences, Engi- medical, including | Insur- and tion Total mathe- chem- meteor- || neering and econom- ance business and Other matics | istry ology agricul- || ics, and adminis-| inven- tural pyS- tration tory Sciences chology Number reporting---------------- 19, 982 |1, 815 848 237 |4,015 131 205 917 502 581 663 Percent citing each course Elementary courses: College algebra--------------- 94 95 96 96 96 94 96 97 82 92 89 Trigonometry---------------- 85 89 94 97 94 79 70 69 50 73 77 Analytic geometry------------ 83 85 92 93 91 82 73 73 47 75 72 Differential and integral calcu- - lus------------------------ 85 95 96 94 92 87 79 87 42 72 74 Foundations: Foundations of mathematics--- 19 25 19 17 18 17 21 7 24 24 22 Set theory------------------- 14 18 18 10 13 18 17 2 8 17 20 Mathematical logic----------- 30 34 25 24 29 32 28 16 29 38 37 Other----------------------- 2 2 2 ------ 2 2 1 5 2 2 4 Algebra and number theory: - - Higher algebra--------------- 40 46 45 36 39 48 37 45 25 31 37 Theory of equations---------- 45 55 56 56 48 44 32 35 21 31 36 Algebra of vectors and matrixes- 49 59 68 61 57 58 40 3 19 40 41 Theory of rings and fields - - - - - 3 5 4 3 3 2 2 (2) 1 3 4 Abstract algebraic structures--- l 3 2 2 1 2 1 (2) (2) | 1 2 Tensor algebra (multilinear) - - - 4 4 11 6 5 1 l (2) 1 2. 3 Theory of groups------------- 6 8 10 4 5 7 5 2 5 4 9 Lattice theory.--------------- 2 2 3 1 1. 6 1 ------ 1 1 1 Elementary number theory - - - - 16 23 19 15 16 25 12 8 13 15 17 Algebraic number theory ------ 5 7 4 3 5 2 3 1 6 5 5 Analytic number theory.------- 4 6 3 2 4 7 1 1 6 5 4 Other----------------------- l 1 1 1 1 1 (2) 3 (2) 1 1 Analysis: . Differential equations--------- 60 63 82 65 73 56 28 25 21 42 48 Advanced calculus------------ 47 49 70 53 54 48 32 26 12 30 39 Vector analysis--------------- 36 41 58 54 45 19 7 1 8 19 28 Partial differential equations--- 32 33 57 41 41 26 9 7 8 17 21 Integral equations------------ 17 19 29 21 21 16 4 7 5 10 12 Calculus of finite differences - - - 24 18 27 23 21 23 17 68 6 13 19 Calculus of variations--------- 11 11 21 9 13 11 6 2 2 8 9 Series and summability-------- 19 20 26 22 19 19 11 21 7 12 16 Functional analysis----------- 6 8 10 5 6 5 4 1 7 5 7 Linear spaces & operator theory- 4 6 9 3 5 2 2 (2) 1 2 3 Hilbert spaces---------------- 1 2 5 1 1 ------- I (2) (2) (2) 1 Banach spaces--------------- 1 1 2 (2) (*) ------- 1 ------------------ 1 Functions of real variables_____ 16 19 29 19 19 23 8 2 4 9 16 Functions of complex variables- 22 22 38 24 29 19 8 1 3 9 17 Algebraic functions----------- 8 10 10 6 8 6 2 4 6 7 6 Special functions------------- 5 6 12 10 5 3 1 (2) 1 2 2 Orthogonal functions----- = = ** = - 11 12 20 14 13 17 4 (2) 1 6 9 Integral transforms----------- 9 8 19 8 13 11 3 (2) 1 3 5 Harmonic analysis------------ 10 9 17 20 14 5 2 ------ 2 2 5 Potential theory.-------------- - 6 5 16 15 7 1 (*) ------ 1 1 3 Lebesgue measure and inte- s gration-------------------- 3 4 6 4 3 5 5 l------ (2) | 2 | 3 Theory of conformal mapping--| 5 5 10 10 7 2 1 ------ (2) 1 3 Numerical analysis----------- ... 39 48 51 48 || 45 34 15 4 20 29 35 Other----------------------- 1 1 1 3. 2 2 (2) 2 1 (2) 1 See footnotes at end of table. 49 TABLE A–25.-Mathematics courses cited by respondents as minimum requirements for their positions, by chief subject matter source of problems, 1960–Continued Subject matter Source of problems Biologi- Social Courses cited Earth cal, Sciences Office | Produc- Basic | Physics, sciences, Engi- medical, including | Insur- and tion Total mathe- chem- meteor- neering and, econom- ance business! and Other matics istry ology agricul- || ics, and adminis- inven- tural pyS- tration tory Sciences | chology Percent citing each course Topology: Algebraic (combinatorial) to- pology-------------------- 2 3 3 2 2 2 4 ------ 1 2 3 Point set topology------------ 3 5 3 1 3 3 4 ------ (2) 2 3 Topological groups----------- 1 1 1 (2) 1. 2 (*) ------ (2) (2) (2) Homotopy groups------------ (2) (2) (2) (2) (*) -------|-------|------|------ (*) ------ Homology theory.------------- (2) (2) 1 ------ (*) -------|-------|------|------|------ (2) Lie groups------------------- (2) (2) 1 ------ (*) ------- ! ------------------ (2) Lie algebras----------------- (2) (2) 1 ------ (*) ------- (*) ------|------|------|------ Other----------------------- (2) (2) (*) ------ (*) -------|-------|------|------|------|------ Geometry: Advanced analytic geometry--- 16 20 21 23 18 5 5 3 3 10 17 Advanced Euclidian geometry-- 5 6 7. 6 5 2 2 1 1 3 5 Non-Euclidian geometry.-- - - - - 2 3 3 2 2 2 1 (2) 1 2 3 Projective geometry - - - - - - - - - - 6 8 6 13 6 4 1 1 1. 2 5 Algebraic geometry----------- 3 5 3 3 4 1 1 1. 1 3 3 Differential geometry--------- 6 8 10 10 6 1 (*) ------ 1. 1 5 Other----------------------- (2) 1 (*) ------|------|-------|------- (*) ------ (2) (2) Probability and statistics: Elementary statistics--------- 58 51 54 47 51 89 87 80 74 77 6] Mathematical statistics- - - - - - - 45 40 39 40 38 84 73 66 52 61 47 Statistical inference----------- 19 13 16 8 15 74 55 18 25 39 26 Probability theory--- - - - - - - - - - 45 34 39 30 38 80 67 81 42 60 50 Information theory.----------- 10 8 9 8 11 20 17 2 11 11 14 Stochastic processes----------- 11 9 12 6 12 31 19 2 6 19 14 Statistical decision theory - - - - - 11 8 9 3 10 31 29 5 16 28 15 Theory of games------------- 11 11 10 7 10 17 22 8 13 22 | 18 Correlation analysis----------- 23 15 20 18 19 66 60 | 16 33 45 29 Sampling theory-------------- 24 16 16 13 17 73 62 30 46 46 32 Design and analysis of experi- ments--------------------- 19 12 21 10 20 80 42 5 13 34 25 Nonparametric methods------- 8 5 8 1 8 57 24 1 5 16 11 Other----------------------- 2 1 2 (2) 1 8 3 10 2 4 4 Miscellaneous: Fluid dynamics-------------- 11 5 21 9 18 2 1 ------|. (2) 2 4 Electrodynamics------------- 6 3 16 8 8 1 (*) ------ 1 1. 2 Elasticity theory.------------- 5 2 11 10 9 2 (2) (2) (2) 1 2 Magneto-hydrodynamics------ 2 1 7 1. 2 1 -------|------|------ (2) 1 Linear vibrations------------- 9 4 15 8 15 --------------|------ (2) 1 2 Nonlinear vibrations---------- 5 3 11 5 9 --------------|------ (2) (2) 2 Mathematical theory of com- puters and control devices--- 25 29 23 28 29 21 13 10 21 23 25 Programming of high-speed digital computers----------- 42 48 46 52 46 29 30 16 32 47 44 Linear programming---------- 22 23 18 19 21 23 28 4 33 46 23 Mathematical methods in oper- - ations analysis------------- 21 20 17 16 20 19 26 4 34 42 28 Other----------------------- 6 3 7 5 5 2 2 21 2 2 6 No courses reported- - - ------- 2 2 1 (2) 1 1 1 2 || 4 1 3 1 Includes 68 respondents who did not specify their subject matter source of problems. * Less than 0.5 percent. 50 TABLE A-26.-Mathematics courses cited by respondents as minimum requirements for their positions, by chief subject matter source of problems, for persons with a doctor’s degree in mathematics, 1960 Subject matter source of problems Biologi- Social Courses cited Earth Cal, Sciences Office Produc- Basic Physics, Sciences, Engi- medical, including | Insur- and tion Total mathe- chem- meteor- neering and €COIlOrn- ance business and Other maticS istry ology agricul- ics, and adminis- inven- tural tration tory pSy- sciences chology Number reporting---------------- 1 441 || 114 71 9 || 154 14 14 5 4 | 16 38 Percent citing each course Elementary courses: College algebra--------------- 93 91 97 100 92 100 100 100 50 100 87 Trigonometry---------------- 88 87 99 100 87 93 86 60 50 75 84 Analytic geometry------------ 92 91 96 100 90 100 93 100 50 94 87 Differential and integral calcu- lus------------------------ 93 91 97 100 91 100 100 80 50 100 89 Foundations: - *. Foundations of mathematics--- 33 38 37 11 27 29 29 ------|------ 31 45. Set theory------------------- 46 56 39 33 42 36 50 |------ 25 50 53 Mathematical logic----------- 29 34 24 11 27 21 43 |------ 25 38 29 Other----------------------- 3 3 3 ------ 2 7 -------|------------ 6 8 Algebra and number theory: Higher algebra--------------- 60 68 63 56 55 79 57 40 50 50 50. Theory of equations- - - - - - - - - - 62 75 62 56 59 64 43 20 50 44 53 Algebra of vectors and matrixes— 83 89 92 78 81 86 64 e-º º wº: * 50 100 68. Theory of rings and fields- - - - - 13 22 13 |------ 8 ------- 7 ------------ 19 16 Abstract algebraic structures--- 7 12 6 ------ 6 7 -------|------------ 6 5. Tensor algebra (multilinear) - - - 16 19 30 11 14 --------------|------------ 12 8. Theory of groups------------- 26 34 28 ------ 21 14 14 ------|------ 25 39 Lattice theory--------------- 5 9 1 ------ 5 14 14 ------------ 6 3 Elementary number theory.---- 25 35 20 ------ 23 43 21 ------ 25 6 24 Algebraic number theory - - - - - - 4 5 3 ------ 5 -------|------------- 25 6 3. Analytic number theory.------- 2 4 ------|------ 2 -------|-------------|------ 6 sºme sºns me sºme Other----------------------- 1 2 1 ------ 1 -------|-------------------|------ 3. Analysis: Differential equations--------- 81 82 97 78 81 86 71 20 50 69 68. Advanced calculus------------ 82 80 96 89 84 71 79 20 50 69 76 Vector analysis--------------- 62 61 83 67 65 29 21 |------ 25 50 63 Partial differential equations_ _ _ 58 56 86 78 59 64 14 |------ 25 31 39, Integral equations------------ 44 46 62 67 44 43 7 20 ------|------ 26 Calculus of finite differences---- 46 51 51 22 44 50 43 100 25 50 39. Calculus of variations--------- 38 42 51 11 40 14 7 ------ 50 25 29, Series and Summability-------- 42 51 48 33 41 36 29 . 40 |------ 25 34. Functional analysis----------- 27 42 25 11 27 7 21 ------|------ 12 13, Linear spaces & operator theory- 22 34 27 11 18 14 14 ------ 25 12 11 Hilbert Spaces---------------- 13 24 23 11 7 ------- 7 ------|------|------ 3. Banach Spaces--------------- 8 17 10 11 3 7 ------------|------ 3 Eunctions of real variables----- 58 72 73 67 50 43 29 |------ 25 44 50. Functions of complex variables— 68 75 87 89 69 43 43 |------ 25 25 58. Algebraic functions----------- 10 11 15 ------ 8 -------|-------|------|------ 6 11 Special functions------------- 27 32 48 22 27 7 7 ------|------|------ 5. Orthogonal functions---------- 38 44 56 22 34 29 21 ------ 25 25 29. Integral transforms----------- 36 42 56 22 37 29 21 ------|------ 12 11 Harmonic analysis------------ 28 35 35 44 28 14 7 ------|------ 12 16 Potential theory-------------- 24 30 45 33 23 | | |------------------ 8. Lebesgue measure and integra- tion----------------------- 26 37 27 44 24 14 29 ------|------ 12 13 Theory of conformal mapping-- 22 30 31 33 21 | | |------ 25 6 5. Numerical analysis----------- 58 61 69 67 58 71 14 ------ 25 62 42. Other----------------------- 2 3 1 ------ 1 -------|-------|------|------ 6 ------ See footnotes at end of table. 51. TABLE A-26–Mathematics courses cited by respondents as minimum requirements for their positions, by chief subject matter source of problems, for persons with a doctor’s degree in mathematics, 1960–Continued Subject matter Source of problems Biologi- Social Courses cited Earth - cal, Sciences Office | Produc- Basic | Physics, sciences, Engi- medical, including | Insur- and tion Total mathe- chem- meteor- neering and econom- ance business and Other matics istry Ology . agricul- || ics, and adminis-| inven- tural pyS- tration tory Sciences chology Percent citing each course Topology: Algebraic (combinatorial) to- pology-------------------- 12 21 8 ------ 11 ------- 21 ------|------ 6 11 Point set topology------------ 22 38 13 33 18 7 29 ------|------ 12 18 Topological groups----------- 2 6 3 ------ 1 -------------------------------------- Homotopy theory.------------ 1 4 1 ------|------|--------------------|------|------------ Homology theory------------- 1 3 3 --------------------------|------------------|------ Lie groups------------------- 2 2 3 ------ 2 ------- 7 ------------------------ Lie algebra------------------ 1 ------ 3 ------ 1 ------- 7 ------------------------ Other----------------------- (2) 1 --------------------------------------|------------------ Geometry: Advanced analytic geometry--- 31 36 37 56 31 14 21 ------|------ 19 26 Advanced Euclidian geometry-- 11 16 10 22 7 14 7 ------------ 6 18 Non-Euclidian geometry------ 5 6 3 ------ 5 --------------------|------ 6 8 Projective geometry---------- 10 15 10 33 5 7 7 ------------ 6 8 Algebraic geometry----------- 3 4 3 ------ 2 ------- 7 ------------------ 3 Differential geometry--------- 25 34 32 44 20 7 -------------|------ 6 24 Other----------------------- 1 2 ------------ 1 -------------------------------------- Probability and statistics: Elementary statistics---------- 61 49 59 33 63 100 79 100 75 81 68 Mathematical statistics------- 62 54 46 78 67 100 86 80 50 69 68 Statistical inference----------- 35 28 27 11 36 93 79 20 25 38 34 Probability theory.------------ 70 68 59 44 71 100 100 80 50 69 84 Information theory.----------- 25 27 20 11 || 29 36 29 |------ 25 25 18 Stochastic processes----------- 37 39 31 11 36 57 57 ------ 25 44 42 Statistical decision theory - - - - - 23 19 18 ||------ 27 43 43 ------------ 38 21 Theory of games------------- 24 26 18 11 21 14 43 ------|------ 50 34 Correlation analysis----------- 28 22 20 22 26 86 50 20 25 56 34 Sampling theory-------------- 26 20 20 l------ 22. 93 79 20 25 44 32 Design and analysis of experi- ments--------------------- 29 19 23 11 31 93 79 ------|------ 44 32 Nonparametric methods------- 22 16 17 ------ 23 86 50 ------|------ 19 21 Other----------------------- 2 1 1 ------ 1 14 14 20 ------ 6 ------ Miscellaneous: Fluid dynamics-------------- 24 22 48 56 26 -------|-------|------|------|------ 8 Electrodynamics------------- 15 11 38 33 15 --------------------|------------ 5 Elasticity theory.------------- 14 10 28 22 17 --------------------|------|------ 3 Magneto-hydrodynamics------ 5 5 18 11 2 -------|------------------------- 3 Linear vibrations.------------- 24 25 35 22 31 --------------------------|------ 3 Nonlinear vibrations---------- 18 20 30 |------ 23 -------|-------|------|------|------ 3 Mathematical theory of com- - puters and control devices--- 25 25 24 22 29 21 14 ------------ 25 26 Programming of high-Speed digital computers----------- 37 28 42 22 38 29 29 |------ 25 75 53 Linear programming---------- 30 26 28 11 31 29 57 |------ 25 69 32 Mathematical methods in op- - * erations analysis------------ 32 29 17 11 36 29 57 ------ 50 62 45 Other----------------------- 7 5 8 11 8 -------|------- 40 25 ------ 11 No courses reported.---------- 2 1 1 ------ 2 --------------|------ 25 ------ 5 1 Includes 2 respondents who did not report Source of problems. 2 Less than 0.5 percent. 52 TABLE A–27.-Mathematics courses cited by respondents as minimum requirements for their positions source of problems, for persons with a master's degree in mathematics, 1960 , by chief subject matter Subject matter source of problems See footnotes at end of table. Biologi- Social Courses cited Earth cal, Sciences Office | Produc- Basic | Physics, Sciences, Engi- medical, including | Insur- and tion Total mathe- chem- || meteor- | neering and econom- ance business and Other matics istry ology agricul- || ics, and adminis- inven- tural pyS- tration tory Sciences chology Number reporting---------------- 1 1,580 288 || 153 33 620 34 34 155 39 74 133 Percent citing each course Elementary courses: College algebra--------------- 95 95 97 94 96 94 97 100 95 97 92 Trigonometry---------------- 88 93 93 100 93 88 74 77 72 78 81 Analytic geometry------------ 88 90 95 100 92 88 76 81 74 85 77 Differential and integral cal- * culus---------------------- 93 93 98 97 96 94 91 94. 79 89 83 Foundations: Foundations of mathematics--- 21 29 24 12 2I 21 9 | 5 26 27 25 Set theory------------------- 22 26 26 18 21 26 21 3 18 30 32 Mathematical logic----------- 31 36 24 33 31 38 18 14 33 43 45 Other----------------------- 3 2 4 ------ * -------------- 4 3 3 6 Algebra and number theory: Higher algebra--------------- 44 52 47 48 42 59 44 45 36 34 41 Theory of equations---------- 57 67 69 73 62 53 29 34 38 43 44 Algebra of vectors and ma- trixes--------------------- 61 72 77 82 69 76 59 2 46 58 54 Theory of rings and fields----- 4 7 3 3 4 6 3 ------ 3 5 5 Abstract algebraic structures--- 2 5 1 3 * ------- 6 ------ 3 ------ 2 Tensor algebra (multilinear) --- 4 5 7 6 * !-------------------------- 1 5 Theory of groups------------- 8 14 9 3 7 9 9 ------ 10 3 12 Lattice theory.--------------- 2 4 2 ------ 1. 3 ------------------- 1 2 Elementary number theory.---- 21 26 31 24 22 21 12 7 18 26 18 Algebraic number theory.------ 4 7 7 3 4 3 ------- 1 ------ 4. 6 Analytic number theory.------- 4 6 1. 3 4 3 ------------- 8 3 5 Other----------------------- 2 2 2 3 * !-------------- 5 |------ 1 ------ Analysis: - Differential equations--------- 70 76 91 85 82 65 29 29 46 58 59 Advanced calculus------------ 64 70 85 88 70 65 53 30 26 51 56 Vector analysis--------------- 42 54 58 85 52 18 6 1 15 27 31 Partial differential equations--- 37 45 65 61 43 26 9 6 15 22 23 Integral equations------------ 18 20 26 30 21 15 3 5 10 8 16 Calculus of finite differences--- 36 29 40 48 32 41 24 77 23 27 29 Calculus of variations--------- 15 16 20 12 19 I2 6 1 8 15 I0 Series and Summability-------- 26 30 39 36 25 21 12 19 26 16 19 Functional analysis----------- 9 12 10 6 10 3 6 1. 5 7 8 Linear spaces & operator theory- 6 9 7 6 * -------------------- 3 1 5 Hilbert Spaces---------------- 1 3 3 |------ ! -------|------------------------- 2 Banach Spaces--------------- 1 2 2 ------ ! -------------------------------- 1 Functions of real variables----- 27 31 44 33 31 32 12 1 13 16 26 Functions of complex variables- 36 41 49 45 46 38 15 1 15 19 24 Algebraic functions----------- 7 11 9 6 7 3 ------- 3 5 4 5 Special functions------------- 7 11 14 6 9 6 ------- 1 ------ 1 Orthogonal functions---------- 16 16 24 30 20 18 3 ------|------ 12 13 Integral transforms----------- 14 13 22 15 19 12 3 ------ 3 4 11 Harmonic analysis------------ 12 12 14 33 17 6 3 ------------ 4 8 Potential theory.-------------- 8 8 14 36 10 -------|------------------------- 2 Lebesgue measure and integra- - - tion----------------------- 6 8 7 6 6 12 6 ------------ 7 9 Theory and conformal mapping- 6 7 10 21 7 6 3 ------ 3 1 4 Numerical analysis----------- 51 65 65 67 58 41 21 4 44 38 45 Other----------------------- 1. 3 9 1 3 ------- 3 ------------------ 53 TABLE A–27.-Mathematics courses cited by respondents as minimum requirements for their positions, by chief subject matter source of problems, for persons with a master's degree in mathematics, 1960–Continued Subject matter source of problems Biologi- Social Courses cited Earth cal, Sciences Office Produc- Basic | Physics, sciences, Engi- || medical, including | Insur- and tion Total mathe- chem- meteor- neering and econom- ance business and Other matics istry ology agricul- ics, and adminis- inven- tural pyS- tration tory sciences | chology . Percent citing each course Topology: - Algebraic (combinatorial) to- pology-------------------- 3 4 4 ------ 3 3 3 ------ 5 4 5 Point Set topology------------ 5 8 7 ------ 4 9 3 ------ 3 5 4 Topological groups----------- (2) 1 1 ------ (2) 6 -------|------|------|------ 1 Homotopy theory.------------ (*) ------|------|------ (*) -------|-------|------|------|------|------ Homology theory------------- (2) 1 1 ------ (*) -------|-------|------|------|------|------ Lie groups------------------- (*) ------ ! ------ (*) -------|-------|------|------|------|------ Lie algebras----------------- (*) ------ 1 ------ (*) -------|-------|------|------|------|------ Other----------------------- (*) ------ 1 ------ (*) -------|-------|------|------|------|------ Geometry: Advanced analytic geometry--- 19 25 27 45 20 6 9 2 3 11 23 Advanced Euclidian geometry-- 6 5 11 12 6 3 3 ------ 3 4 7 Non-Euclidian geometry------ 3 2 8 3 2 3 3 ------ 3 8 2 Projective geometry---------- 7 9 10 27 7 6 3 ||------ 3 5 7 Algebraic geometry---------- 3 7 2 6 2 -------------- 1 ------ 4 5 Differential geometry__ _ _ _ _ _ _ 9 16 15 27 10 --------------|------|------ 1 5 eſ----------------------- 1 1 1 ------ 1 -------|-------|------|------|------|------ Probability and statistics: Elementary statistics-- - - - - - - - 67 59 59 73 62 88 88 89 72 84 68 Mathematical statistics- - - - - - 59 54 52 58 55 97 82 75 49 70 58 Statistical inference----------- 29 22 24 12 28 82 71 21 23 59 34 Probability theory.------------ 58 48 47 55 53 94 74 90 49 74 59 Information theory.-- - - - - - - - - - 14 12 10 18 17 15 12 2 8 18 18 Stochastic processes----------- 16 14 14 15 19 38 21 11 10 27 18 Statistical decision theory – - - - 15 12 10 6 15 41 38 3 13 41 15 Theory of games------------- 14 15 9 3 12 21 29 6 18 31 21 Correlation analysis----------- 29 22 24 27 28 68 65 16 26 62 30 Sampling theory.-------------- 27 17 18 27 22 74 74 32 18 57 35 Design and analysis of experi- ments--------------------- 26 19 25 15 27 82 44 5 18 55 32 Nonparametric methods------- 14 8 12 3 15 71 26 2 10 30 17 Other----------------------- 4 1 2 ------ 2 18 3 17 8 7 6 Miscellaneous: Fluid dynamics-------------- 9 5 15 12 14 6 ------------------- 1 5 Electrodynamics------------- 4 2 12 12 6 3 -------|------------ 1 2 Elasticity theory.------------- 5 3 6 24 7 3 -------|------|------ 1 2 Magneto-hydrodynamics- - - - - - 1 ------ 5 ------ 1 -------|-------|------------|------|------ Linear vibrations------------- 8 3 14 15 13 -------|------------------- 5 2 Nonlinear vibrations---------- 4 1. 10 6 7 -------|------------------------- 2 Mathematical theory of com- puters and control devices--- 25 33 27 21 27 24 6 10 33 18 29 Programming of high-Speed digital computers----------- 47 53 53 58 48 35 35 18 56 50 50 Linear programming---------- 25 24 21 24 23 26 32 6 49 59 32 Mathematical methods in op- erations analysis------------ 25 25 18 24 27 21 15 5 41 46 33 Other----------------------- 9 7 12 9 7 3 3 26 3 5 5 No courses reported- - - - ------ 1 2 ------------ 1 ------- 3 ------ 5 ------ 2 1 Includes 17 respondents who did not specify their chief subject matter Source of problems. * Less than 0.5 percent. 54 TABLE A-28.-Mathematics courses cited by respondents as minimum requirements for their positions, by chief subject matter Source of problems, for persons with a bachelor’s degree in mathematics, 1960 Subject matter source of problems Biologi- Social Courses cited Earth |. Cal, Sciences Office | Produc- Basic | Physics, sciences, Engi- medical, including | Insur- and tion Total mathe- || chem- meteor- neering and €COI101m- ance business! and Other matics | istry ology agricul- ics, and adminis-l inven- tural pyS- tration tory Sciences chology Number reporting---------------- 14, 290 990 326 125 |1, 57.1 26 56 567 122 194 279 Percent citing each course Elementary courses: College algebra.--------------- 95 96 95 95 97 92 96 96 84 93 89 Trigonometry---------------- 86 91 95 95 95 73 70 63 58 75 78 Analytic geometry------------ 83 87 92 91 90 77 68 70 56 73 73 Differential and integral cal- culus---------------------- 86 88 95 91 92 92 75 86 48 68 76 Foundations: Foundations of mathematics_ _ _ 17 23 17 17 17 12 18 7 17 27 17 Set theory------------------- 11 14 12 7 12 15 14 3 10 19 16 Mathematical logic----------- 32 35 29 29 35 27 29 16 36 46 40 Other----------------------- 2 2 2 ------ 1 4 5 5 l 3 3 Algebra and number theory: - Higher algebra--------------- 41 46 44 33 40 42 46 44 25 38 39 Theory of equations- - - - - - - - - - 50 58 60 53 55 35 48 34 30 38 36 Algebra of vectors and mat- rixes---------------------- 48 58 63 61 58 62 38 3 27 38 37 Theory of rings and fields- - - - - 3 4 3 3 2 ------- 2 (2) 2 4 3 Abstract algebraic structures--- 1 2 2 2 ! ------- 2 (*) ------ 3 1. Tensor algebra (multilinear) - - - 3 3 4 5 4 4 2 (*) ------ 1 1 Theory of groups------------- 5 5 6 6 4 4 4 2. 4 6 6 Lattice theory.--------------- 1 1. 2 1 1 8 ||-------|------------ 2 1 Elementary number theory - - - - 18 22 19 17 19 38 14 9 16 21 17 Algebraic number theory.------ 6 8 4 4 5 4 7 1 6 9 6 Analytic number theory.------- 4 6 5 3 4 ------- 2 1. 2 6 5 Other----------------------- 1 1 ! ------ 1 -------|------- 4 ------ 2 1 Analysis: Differential equations--------- 58 64 77 61 72 58 25 24 25 40 46 Advanced calculus------------ 44 48 59 46 51 46 23 26 20 30 32 Vector analysis--------------- 32 39 49 43 42 19 4 1 11 18 25 Partial differential equations--- 29 32 44 34 37 23 12 7 7 21 19 Integral equations------------ 14 17 17 14 16 12 11 7 4 12 9 Calculus of finite differences – - 22 14 18 18 17 4 20 67 6 14 13 Calculus of variations--------- 7 8 11 6 8 12 4 2 ------ 6 6 Series and Summability-------- 17 17 17 18 17 23 9 21 8 12 14 Functional analysis----------- 3 4 5 2 4 -------|------- (2) 1 5 5 Linear spaces & operator theory- 2 3 4 2 2 -------|------- (2) 1 3 2 Hilbert Spaces---------------- 1 1 2 1. 1 -------|-------------|------ 1. 1 Banach spaces--------------- (2) (2) 1 ------ (*) -------|-------|------|------|------ (2) Functions of real variables----- 11 13 16 14 14 5 2 4 9 10 Functions of complex variables- 15 15 21 17 20 ------- 5 2 2 8 11 Algebraic functions----------- 7 10 7 6 7 4 4 3 5 10 Special functions------------- 2 3 5 2 2 ------- 2 ------------ 4 Orthogonal functions---------- 7 8 10 6 8 12 2 (2) 2 7 Integral transforms----------- 4 5 5 5 6 -------------- (*) ------ 3 Harmonic analysis------------ 6 6 8 13 9 4. 2 ------ 3 3 Potential theory.-------------- 2 2 5 6 2 ------- 2 ------ 1 1 Lebesgue measure and integra- tion----------------------- 1 1 2 2 2 ------- 2 ------|------ 2 Theory of conformal mapping-- 2 3 3 5 3 --------------|------------ 3 Numerical analysis----------- 39 48 52 45 49 23 18 4 23 35 3 Other----------------------- 1. 1 1 ------ 2 4 2 2 2 ------ See footnotes at end of table. 55 TABLE A-28-Mathematics courses cited by respondents as minimum requirements for their positions, by chief subject matter source of problems, for persons with a bachelor's degree in mathematics, 1960–Continued Subject matter source of problems Biologi- Social Courses cited IEarth cal, Sciences Office | Produc- Basic Physics, sciences, Engi- medical, including Insur- and tion Total mathe- chem- meteor- neering and eCODOIn- ance business and Other - matics istry ology agricul- || ics, and adminis- inven- tural pyS- tration tory Sciences chology Percent citing each course Topology: Algebraic (combinatorial) to- Pology-------------------- 2 2 2 3 2 --------------|------------ 2 2. Point set topology------------ 1. 2 1 ------ 2 -------|-------|------------ 1 1 Topological groups----------- (2) 1 ------ 1 1 -------|-------|------------ 1 1 Homotopy theory.------------ (*) ------|------|------ (*) -------|-------|------|------ 1 ------ Homology theory------------- (*) ------|------|------ (*) -------|-------|------|------|------|------ Lie group-------------------- (2) (*) ------|------ (*) -------|-------|------|------|------|------ Lie algebras----------------- (2) (*) ------|------ (*) -------|-------|------|------|------|------ Other-----------------------|-------------------------|------|-------------------------------------- Geometry: Advanced analytic geometry--- 15 19 18 20 18 l-------|------- 3 5 12 16. Advanced Euclidian geometry- 4. 6 4 5 4 ------- 4 1 2 5 4. Non-Euclidian geometry---li- 3 3 2 4 3 -------|------- (2) 3 2 4 Projective geometry---------- 6 9 5 13 7 4 ------- 1 3 3 5. Algebraic geometry----------- 3 4 3 2 4 -------|------- 1 2 4 3 Differential geometry--------- 4 5 5 6 5 -------|-------|------------ 2 2 Other----------------------- I 1 1 ------ (*) -------|------- (*) ------|------ 1. Probability and statistics: Elementry statistics---------- 54 48 48 45 49 92 86 77 53 71 47 Mathematical statistics------- 41 35 35 34 37 85 75 64 39 54 34 Statistical inference----------- 13 8 9 6 12 69 36 16 12 28 16 Probability theory.------------ 38 26 28 20 33 73 54 79 35 48 36 Information theory.----------- 6 5 5 6 7 12 12 2 8 10 8 Stochastic processes----------- 6 4 6 2 7 19 9 2 7 13 8 Statistical decision theory.----- 7 5 4 2 7 31 18 5 11 25 9 Theory of games------------- 9 8 6 6 7 23 20 7 15 16 11 Correlation analysis----------- 16 11 12 12 15 73 45 15 24 34 20 Sampling theory-------------- 17 11 11 10 14 69 52 27 22 37 21 Design and analysis of experi- ments--------------------- 13 9 13 7 15 88 41 5 12 24 17 Nonparametric methods------- 4 3 4 2 5 50 11 1 2 13 5 Other----------------------- 2 1 2 ------ 1 8 ------- 9 1 2 3 Miscellaneous: Fluid dynamics-------------- 5 3 8 2 10 -------|-------|------ 1 2 3 Electrodynamics------------- 3 2 6 2 5 -------|-------|------ 2 1 3 Elasticity theory.------------- 2 I 4 2 4 --------------|------ 1 1 1 Magneto-hydrodynamics------ 1 1 1 ------ 1 -------|-------|------|------ 1. 1. Linear vibrations------------- 3 2 6 I 6 -------------------- 1 2 1 Nonlinear vibrations---------- 2 2 3 1 4 --------------|------ 1. 1 1 Mathematical theory of com- puters and control devices-- 26 30 23 34 30 19 18 10 25 29 24 Programming of high-speed digital computers----------- 45 50 49 59 53 42 32 16 39 50 43 Linear programming---------- 19 22 16 21 21 15 23 4 30 41 17 Mathematical methods in op- erations analysis------------ 16 16 13 14 16 12 25 3 30 35 20 Other----------------------- 6 3 4 3 5 4 2 20 1 1 5 No courses reported---------- 2 2 2 1. 1 ------- 2 2 9 3 4 * Includes 34 respondents who did not specify subject matter source of problems. * Less than 0.5 percent. 56 wnder their supervision, by supervisor's subject matter source of problems, 1960 TABLE A-29.-Mathematics courses cited by supervisors as minimum requirements for a typical bachelor's degree level position Supervisor’s subject matter source of problems See footnotes at end of table. A & Courses cited sºlet Earth Biological, sitº, Office | Produc- matter Basic |Physics, sciences, Engi- medical, including | Insur- and |tion and area,S mathe- chemis- meteor- | neering and agri- econom- ance business inven- | Other matics try Ology cultural ics and admin- tory sciences | psychol- istration Ogy Number reporting---------------- 12, 554 351 | 207 50 | 956 54 57 || 444 || 130 | 122 168 Percent of supervisors citing each course Elementary courses: Algebra--------------------- 94 93 93 82 95 96 98 95 98 92 87 Trigonometry---------------- 85 89 89 84 93 85 79 67 72 79 84 Analytic geometry------------ 84 || 87 87 84 92 81 75 70 68 82 S2 Differential and integral calculus- 86 85 90 84 93 91 79 84 65 78 80 Foundations: Foundations of mathematics--- 15 20 15 8 15 9 26 4 33 21 18 Set theory------------------- 7 9 7 6 6 2 16 (2) 18 13 10 Mathematical logic----------- 18 22 14 14 17 17 21 9 35 23 28 Other----------------------- 2 2 2 ------ 1 -------------- 4 1 2 2 Algebra and number theory: Higher algebra--------------- 34 37 38 20 30 26 28 41 39 29 33 Theory of equations---------- 36 47 44 34 39 31 23 23 31 33 30 Algebra of vectors and matrixes- 35 44 57 34 43 41 33 2 25 39 34 Theory of rings and fields- - - - - 1 2 1 ------ 1 --------------|------ 1 1 1. Abstract algebraic structures--- (3) (*) ------|------ (*) -------|-------|------ 1. 1. 1 Tensor algebra (multilinear) - - - 1 1 2 4 2 -------|-------------|------|------ 1 Theory of groups------------- 2 2 1 ------ 2 2 4 1 7 2 3 Lattice theory--------------- 1 ------ 1 ------ 1 2 2 ------ 1 ------ 1 Elementary number theory.---- 10 12 12 14 10 17 11 4. 21 11 11 Algebraic number theory.------ 2 4 (*) ------ 2 4 2 1 7 6 2 Analytic number theory.------- 2 4 ------------ 1. 4 ------- 1 6 4 2 Other----------------------- 1. 1 ------ 2 1 -------------- 3 1 2 1 Analysis: Differential equations--------- 52 57 71 52 69 37 28 17 33 47 46 Advanced calculus------------ 37 39 56 52 43 37 25 18 19 33 37 Vector analysis--------------- 22 30 38 34 29 7 5 (2) 13 14 22 Partial differential equations_ _ _ 18 16 33 20 24 7 11 4 15 16 13 Integral equations------------ 7 8 8 10 9 9 5 4. 6 7 5 Calculus of finite differences--- 18 6 12 10 11 2 9 55 15 9 13 Calculus of variations--------- 4 3 6 4 4 2 4 2 5 6 4 Series and Summability-------- 9 7 8 8 8 4 9 12 11 8 7 Functional analysis----------- 3 3 1 ------ 3 2 4 1. 8 4 2 Linear spaces & operator theory-------------------- 1 1 (*) ------ 2 ------- 2 ------ 1 1 1 Hilbert Spaces---------------- (*) ------|------|------------|------- 2 ------------------|------ Banach spaces--------------- (*) ------|------|------|------|------- 2 ------|------|------------ Functions of real variables----- 6 7 11 2 7 6 5 I 8 2 6 Functions of complex variables— 10 10 17 6 15 2 4 1 5 5 5 Algebraic functions----------- 4 4 4 2 3 4 ------- 4. 7 6 2 Special functions------------- 1 1 3 ------ 1 -------------- (2) 2 2 ------ Orthogonal functions---------- 3 2 8 4 3 ------- 4 ------ 1 2 3 Integral transforms----------- 3 2 4 4 5 ------- 2 ------ 1 ------ 1 Harmonic analysis.------------ 4 1 6 6 6 2 2 ------ 2 1 2 Potential theory-------------- 1 (2) 4 4 1 ------- 2 ------ 2 2 ------ Levesgue IſleaSure and integration---------------- (*) ------ 1 ------ ! ------- 4 ------------ 2 ------ Theory of conformal mapping-- i 2 3 2 2 -------|-------------|------|------------ Numerical analysis.----------- 27 36 34 32 36 22 11 2 22 21 26 Other----------------------- 1 l 1 4 1 -------------- 2 1 ------ 1 57 TABLE A-29–Mathematics courses cited by supervisors as minimum requirements for a typical bachelor's degree level position wnder their supervision, by supervisor's subject matter source of problems, 1960–Continued Supervisor’s subject matter source of problems All Social Courses Subject Earth Biological, Sciences, Office | Produc- Imatter Basic |Physics, sciences, Engi- medical, including | Insur- and |tion and 8TeaS mathe- chemis- || meteor- neering and agri- econom- ance business inven- || Other Imatics try Ology cultural ics and admin- tory Sciences psychol- istration Ogy Percent of Supervisors citing each course Topology: Algebraic (combinatoral) to- Pology-------------------- 1 1 (*) ------ 1 ------- 2 ------ 2 I 1 Point set topology------------ 1 (2) (*) ------ 1 ------- 2 ------ 2 ------ 1 Topological groups----------- (*) ------|------|------ (*) -------|-------|------ 1 ------------ Homotopy theory------------|-------|------|------|------|------|-------|-------|------|------|------|------ Homology theory-------------|-------------|------|------|------|-------|-------|------------|------------ Lie groups------------------- (*) ------|------|------|------|------- 2 ------|------------------ Lie algebras----------------- (*) ------|------|------|------|------- 2 ------------|------------ 9thºr-------------------------------------------------------------------------------------------- Geometry: Advanced analytic geometry--- 9 14 12 14 11 2 2 2 5 11 10 Advanced Euclidian geometry- 2 1. 3 4 2 -------------- (2) 2 2 2 Non-Euclidian geometry - - - - - - 1 1 1 ------ 1 -------------------- 2 1 1 Projective geometry - - -------- 3 4 3 2 4 2 ------------- 2 ------ 2 Algebraic geometry----------- 1. 2 (*) |------ 2 -------------- (*) ------ 2 1. Differential geometry--------- 2 4 ! ------ 3 -------------------- 2 ------ 1 Other----------------------- (2) 1 ------|------ (*) -------|-------|------|------|------ 1 Probability and statistics: - Elementary statistics--------- 56 44 44 44 50 87 84 64 85 70 61 Mathematical statistics - - - ---- 36 26 26 22 27 59 65 52 60 52 40 Statistical inference----------- 12 10 9 2 9 48 40 10 25 24 14 Probability theory------------ 38 22 28 16 30 48 58 65 51 53 43 Information theory.----------- 5 4 3 4 5 6 11 1 8 6 9 Stochastic processes----------- 4 3 4 2 4 4 11 1 7 12 5 Statistical decision theory - - - - - 5 4 5 ------ 4 4 14 1 16 13 9 Theory of games------------- 6 5 4 2 5 4 11 4 16 11 9 Correlation analysis----------- 15 8 13 2 11 48 53 8 38 31 19 Sampling theory.-------------- 16 9 10 4 9 48 56 17 47 31 19 Design and analysis of experi- IneſºtS--------------------- 12 7 13 4 13 59 23 1 14 22 18 Nonparametric methods------- 4 2 3 ------ 3 15 12 (2) 4 11 8 her----------------------- 3 (2) ! ------ 1 6 2 10 2 3 4 Miscellaneous: Fluid dynamics-------------- 5 2 7 ------ 10 -------------------- 1 ------ 1 Electrodynamics------------- 2 1 6 ------ 4 2 ------------- 2 ------ 1 Elasticity theory.------------- 2 1 4 2 4 -------------------------------- 1 Magneto-hydrodynamics------ (*) ------ 1 ------ 1 -------|-------|------|------------------ Linear vibrations------------- 4. 1 4 ------ 9 -------------------- ! ------------ Nonlinear vibrations---------- 1 1 1 ------ 3 -------|------------- 1 ------------ Mathematical theory of com- puters and control devices--- 14 17 13 8 17 6 12 3 24 12 21 Programming of high-speed digital computers----------- 29 36 27 30 33 19 26 7 42 30 39 Linear programming---------- 11 11 4 10 11 4 12 2 38 33 13 Mathematical methods in oper- ations analysis------------- 11 11 6 4 11 4 14 1 39 31 17 Other----------------------- 5 3 3 4 4 ------- 2 16 1 1 4 No courses reported.---------- (*) ------|------|------ (*) -------|------- (*) ------|------|------ * Includes 15 who did not report source of problems. * Less than 0.5 percent. 58 TABLE A–30.-Mathematics courses cited by supervisors as minimum requirements for a typical bachelor's degree level position wnder their supervision, by mathematical field used most by supervisors, 1960 Mathematical field used most Courses cited Com- |Numerical ; Prob- || Actuar- Algebra, puter analysis, including ability ial Logic, including|Topology Total tech- theory of fluid | Analysis and mathe- theory theory and Other niques | computa- |dynamics statistics| matics of Sets Of geometry tion electro- number magnetics Number reporting---------------- 12, 554 696 247 184 || 398 || 464 || 416 13 60 14 41 Percent citing each course Elementary courses: . Algebra--------------------- 94 96 96 94 90 92 95 85 100 71 93 Trigonometry---------------- 85 91 95 95 81 81 68 77 100 79 83 Analytic geometry------------ 84 89 93 93 78 80 73 77 | 100 71 85 Differential and integral calcu- - lus------------------------ 86 89 95 93 77 83 85 100 98 71 78 Foundations: Foundations of mathematics--- 15 20 15 11 20 15 4 23 23 14 17 Set theory------------------- 7 10 7 ------- 7 9 (?) 15 10 50 12 Mathematical logic----------- 18 29 14 9 18 14 7 62 17 14 20 Other----------------------- 2 1. 2 3 2 1 4 8 ------------------ Algebra and number theory: Higher algebra--------------- 34 33 34 28 31 64 1 31 48 14 46 Theory of equations_ _ _ _ _ _ _ _ _ _ 36 40 53 30 39 31 24 15 37 ------ 37 Algebra of vectors and matrixes- 35 46 49 45 37 33 1 38 52 14 37 Theory of rings and fields- - - - - 1 1 1. 1 1 (*) ------|------|------|------|------ Abstract algebraic structures---| (?) ! ------- 1 ------------|------------ ----2 ||------------ Tensor algebra (multilinear) - - - 1. 1. (2) 3 3 1 ------|------ 2 ------ 2 Theory of groups------------- 2 3 ------- 2 2 3 1 8 3 |------ 5 Lattice theory.--------------- 1. 1 2 1 1. 1 ------ 8 ------|------------ Elementary number theory_ _ _ _ 10 18 9 7 9 9 4 8 10 |------ 12 Algebraic number theory - - - - - - 2 4 2 1 3 3 1 ------ 3 ------|------ Analytic number theory_-_ _ _ _ _ 2 2 1. 1 4 2 1 ------------------ 2 Other----------------------- 1 1 (*) ------- 2 1 3 ------|------------ 2 Analysis: Differential equations_ _ _ _ _ _ _ _ _ 52 60 74 80 60 42 18 46 63 36 56 Advanced calculus------------ 37 37 51 48 38 34 19 31 42 29 39 Vector analysis--------------- 22 24 32 48 31 14 |------ 8 33 29 17 Partial differential equations_ _ _ 18 22 19 34 24 11 5 8 18 7 22 Integral equations------------ 7 9 5 11 9 5 4 ------ 15 |------ 5 Calculus of finite differences_ _ _ 18 14 12 12 7 8 57 23 5 |------ 7 Calculus of variations--------- 4 3 2 7 6 5 (*) ------ 2 ------ 2 Series and summability_____ _ _ _ 9 8 8 9 8 7 12 23 8 ------ 5 Functional analysis----------- 3 2 (2) 3 7 3 1 ------ 2 ------ 5 Linear spaces and operator theory-------------------- 1 ------|------- 2 2 1 ------ 62 ------------ 2 Hilbert Spaces---------------- (*) ------|-------|-------|------ (*) ------|------|------|------|------ Banach spaces--------------- (*) ------|-------|-------|------ (*) ------|------|------|------|------ Functions of real variables----- 6 6 6 11 8 5 1 ------ 8 ------ 17 Functions of complex variables- 10 9 14 23 15 7 1 ------ 7 ------ 17 Algebraic functions----------- 4 3 4 3 5 4 4 ------ 3 21 ------ Special functions------------- 1 1 (2) 3 3 (?) (*) ------|------|------|------ Orthogonal functions---------- 3 3 2 4 4 4 ------------|------|------|------ Integral transforms----------- 3 2 1 12 5 3 ------|------|------------ 5 Harmonic analysis--- - - - - - - - - - 4 4 3 10 5 3 ------|------------------------ Potential theory.-------------- 1 1 ------- 7 2 1 ------|------|------------------ Lebesgue measure and integra- tion----------------------- (2) (*) -------|------- 1 1 ------|------ 1 ------ 2 Theory of conformal mapping__ 1 1 2 8 1 1 ------------------ 7 ------ Numerical analysis - – - - - - - - - - - 27 41 58 28 26 14 3 23 20 14 12 Other----------------------- 1. 2 2 l 1 (2) 2 ------------------------ See footnotes at end of table. 59 TABLE A-30.-Mathematics courses cited by supervisors as minimum requirements for a typical bachelor's degree level position under their supervision, by mathematical field used most by supervisors, 1960–Continued Mathematical field used most Analytical Algebra, Courses cited Com- |Numerical|mechanics Prob- || Actuar- includ- puter analysis, including ability ial Logic, ing Topology Total tech- theory of fluid | Analysis and mathe- theory theory and Other niques | computa- dynamics statistics| matics of Sets of geometry tion electro- number magnetics Percent citing each course Topology: Algebraic (combinatorial) to- Pology-------------------- 1 1 (*) ------- 1 1 ------|------------------ 5 Point set topology------------ 1 -------------------- 1 (*) ------ 8 ------------ 2 Topological groups----------- (*) ------|--------------|------------|------------ 2 7 ------ Homotopy theory------------|-------|--------------------|------|------|------|------|------|------------ Homology theory-------------|-------|------|-------|-------------------|------------|------|------------ Lie groups------------------- (*) ------|-------|-------|------ (*) ------|------|------|------|------ Lie algebras----------------- (*) ------|-------|-------|------ (*) ------|------|------|------|------ 9*--------------------------------------------------|------------------------------------------ Geometry: Advanced analytic geometry--- 9 9 11 15 15 6 2 8 12 14 17 Advanced Euclidian geometry- 2 2 2 2 2 2 (*) |------ 5 ------ 2 Non-Euclidian geometry--- I - 1 1 1 1 1 (*) ------|------|------|------|------ Projective geometry---------- 3 3 3 5 4 ! ------ 15 2 14 2 Algebraic geometry----------- 1 1. 1. 3 3 1 | (*) ------ 2 7 ------ Differential geometry--------- 2 2 2 2 3 2 ------|------ 2 7 2 Other----------------------- (2) (2) 1 1 (*) ------|------|------|------|------|------ Probability and statistics: Elementary statistics--------- 56 46 45 35 57 81 66 || 38 50 7 54 Mathematical statistics------- 36 23 22 14 37 61 53 8 25 ||------ 32 Statistical inference----------- 12 4 4 2 14 35 10 ------ 10 |------ 10 Probability theory.----------- 38 22 19 20 35 61 67 31 32 ||------ 39 Information theory.----------- 5 5 2 3 7 7 1 15 5 ------ 5 Stochastic processes----------- 4 3 2 2 5 10 1 ------ 2 ------ 2 Statistical decision theory.----- 5 3 2 1 8 13 1 ------ 3 ------ 5 Theory of games------------- 6 6 2 2 8 9 4 ------ 5 ------ 10 Correlation analysis----------- 15 9 4 3 18 38 9 |------ 7 ------ 10 Sampling theory-------------- 16 6 3 4 19 39 18 l------ 10 7 12 Design and analysis of experi- ments--------------------- 12 6 6 5 11 39 1 8 7 ------|------ Nonparametric methods------- 4 1. 3 1 4 14 1 ------ 2 ------|------ Other----------------------- 3 (2) (*) ------- 2 6 11 ------------|------|------ Miscellaneous: - . Fluid dynamics-------------- 5 4 3 25 7 1 ------------ 13 |------ 2 Electrodynamics------------- 2 2 2 10 3 1 ------------ 5 ------ 5 Elasticity theory.------------- 2 1 1 12 5 (*) ------|------ 7 ------ 2 Magneto-hydrodynamics------ (2) (*) ------- 3 1 ------|------|------ 2 ------|------ Linear vibrations------------- 4 2 3 22 7 ! ------------------------ 7 Nonlinear vibrations---------- 1 1 (2) 10 2 (*) ------|------|------|------ 7 Mathematical theory of com- puters and control devices--- 14 24 15 12 15 7 3 46 13 l------ 20 Programming of high-speed digital computers----------- 29 53 43 20 23 14 5 54 22 43 24 Linear programming---------- 11 17 6 5 13 13 2 8 15 7 17 Mathematical methods in op- erations analysis------------ 11 13 7 7 18 15 1 8 10 ------ 17 Other----------------------- 5 3 4 5 3 2 16 |------ 3 ------ 10 No courses reported---------- (2) (2) (*) -------|------ (*) ------|------|------|------ - - - - * * * Includes 21 respondents who did not specify mathematical field used most. * Less than 0.5 percent. 60 which they work, 1960 TABLE A-31.-Minimum course requirements cited by persons in mathematical employment by type of organizational unit in Type of organizational unit See footnotes at end of table. Courses cited All or- Com- || Mathe- || Statis- Engi- Opera- |Research] General | Produc- || Admin- ganiza- | puting matics tical neering | tions | labora- techni- tion listration| Other tional | labora- unit unit unit research tory cal staff unit unit units tory unit Number reporting------------------ 19, 982 |2, 288 |1, 229 781 |1, 514 || 791 875 956 167 358 970 Percent citing each course Elementary courses: College algebra----------------- 94 96 97 94 95 96 95 95 94 84 90 Trigonometry------------------ 85 93 93 76 92 86 90 78 73 51 73 Analytic geometry-------------- 83 90 92 74 87 86 88 79 69 49 71 Differential and integral calculus- 85 90 95 78 88 92 93 83 68 46 72 Foundations: Foundations of mathematics----- 19 20 24 20 16 25 18 16 12 22 17 Set theory--------------------- 14 14 20 15 8 25 17 11 3 5 10 Mathematical logic------------- 30 37 29 25 25 36 25 25 25 20 31 Other------------------------- 2. 2 2 2 2 2 2 3 2 2 4 Algebra and number theory: Higher algebra----------------- 40 40 50 41 35 42 44 39 29 28 35 Theory of equations------------ 45 55 62 39 38 44 50 38 32 22 34 Algebra of vectors and matrixes-- 49 60 68 43 46 58 59 28 28 8 32 Theory of rings and fields------- 3 3 5 2 2 4 5 2 1 (2) 2 Abstract algebraic structures----- 1 1. 3 1 1 2 2 1 ------------ 1 Tensor algebra (multilinear) ----- 4 3 6 1 5 3 11 3 1 l 3 Theory of groups--------------- 6 6 9 7 4 7 9 5 1 3 5 Lattice theory.----------------- 2 1. 2 2 2 2 4 1 ------------ 1. Elementary number theory_ _ _ _ _ _ 16 22 21 14 12 18 13 14 9 11 14 Algebraic number theory—l------ 5 6 6 4 4 3 4 4 5 4 4 Analytic number theory.--------- 4. 5 4 4 4 3 4 3 3 6 4 Other------------------------- I 1 1 1 2 1 1 1 2 1. 2 Analysis: Differential equations----------- 60 71 76 42 66 65 74 42 25 16 40 Advanced calculus-------------- 47 51 66 39 43 53 63 35 21 11 32 Vector analysis----------------- 36 41 53 15 43 36 52 22 17 3 23 Partial differential equations----- 32 36 46 19 37 31 50 21 8 5 21 Integral equations-------------- 17 16 25 12 20 16 28 14 4 4 11 Calculus of finite differences----- 24 21 30 18 15 24 28 38 23 19 25 Calculus of variations----------- 11 9 17 7 11 15 21 10 1 2 6 Series and Summability---------- 19 18 28 16 14 20 25 19 10 9 14 Functional analysis------------- 6 5 12 6 5 8 12 6 2 5 5 Linear spaces & operator theory__ 4 3 9 2 4 4 9 3 1 1. 3 Hilbert spaces------------------ 1 1 3 I 1. 1. 4 1 ------ (2) 1 Banach spaces----------------- 1 1 2 (2) (2) (2) 2 (*) ------ (2) 1 Functions of real variables------- 16 15 27 15 14 19 26 13 2 3 10 Functions of complex variables--- 22 22 35 14 25 22 37 16 2 3 12 Algebraic functions------------- 8 7 10 8 9 7 10 8 1. 5 6 Special functions--------------- 5 3 11 3 3 3 12 4 1 1 3 Orthogonal functions------------ 11 9 18 11 10 10 20 8 1 1 5 Integral transforms------------- 9 6 16 5 12 8 20 6 I 1. 4 Harmonic analysis------------ -- 10 9 16 4 13 8 19 8 2 1. 4 Potential theory---------------- 6 3 11 1. 6 3 16 6 ------ 1 3 Lebesgue measure and integra- tion------------------------- 3 2 7 5 1 6 7 2 ------ 1 2 Theory of conformal mapping---- 5 4 10 1 7 4 12 5 1 1 2 Numerical analysis------------- 39 60 57 25 30 35 40 22 10 10 25 Other------------------------- 1 1 1 1. 2 1 2 2 ------ 1 2 61 TABLE A-31.-Minimum course requirements cited by persons in mathematical employment by type of organizational unit in which they work, 1960–Continued Type of organizational unit Mathe- | Statis- Courses cited All Or- Com- Engi- || Opera- |Research General Produc- Admin- ganiza- | puting matics | tical neering tions, labora- techni- | tion istration | Other tional labora- unit unit unit research tory | Cal staff unit Unit, units tory unit Percent citing each course Topology: Algebraic (combinatorial) topol- 08X------------------------- 2 2 3 2 2 4 4 2 ------ 1 2 Point set topology-------------- 3 2 6 1 1 5 5 1 1 1 2 Topological groups------------- 1 1 1 (2) 1 1 1 ------|------ 1 (?) Homotopy theory.-------------- (2) (2) (*) ------ (2) (2) (*) ------|------|------ (2) Homology theory.--------------- (2) (2) (*) ------ (2) (2) (*) ------|------ (2) (2) Lie groups--------------------- (2) (2) 1 (2) (*) ------ (2) (*) ------ (2) (2) Lie algebras------------------- (2) (2) (2) (2) (*) ------ (*) ------|------|------ (2) Other------------------------- (*) ------ (*) ------|------|------|------|------|------|------|------ Geometry: Advanced analytic geometry----- 16 16 24 8 18 17 21 11 10 2 10 Advanced Euclidian geometry---- 5 5 8 3 4 5 6 3 1 1 4 Non-Euclidian geometry-------- 2 3 3 2 2 2 2 1 2 1 2 Projective geometry------------ 6 5 9 3 7 5 8 4 3 1 4 Algebraic geometry------------- 3 3 5 1 4 3 4 2 1 1 2 Differential geometry------------ 6 5 12 2 5 5 12 4 ------ 1. 3 Other------------------------- (2) (2) 1 (2) 1 (2) 1 (2) 2 ------ (2) Probability and statistics: Elementary statistics----------- 58 46 54 91 43 77 57 69 51 76 64 Mathematical statistics_ _ _ _ _ _ _ _ _ 45 31 46 78 28 67 44 55 34 54 48 Statistical inference------------- 19 6 14 63 9 41 17 22 11 21 20 Probability theory.-------------- 45 27 39 77 30 73 44 61 38 49 50 Information theory.------------- 10 6 9 12 9 21 13 10 2 8 8 Stochastic processes------------- 11 5 11 22 7 32 15 8 3 3 7 Statistical decision theory - - - - - - - 11 4 7 32 6 28 10 12 4 9 12 Theory of games--------------- 11 8 9 15 6 35 9 12 5 9 11 Correlation analysis------------- 23 12 16 63 13 41 21 24 17 29 22 Sampling theory.---------------- 24 8 13 66 14 41 20 31 19 44 29 Design and analysis of experi- ments----------------------- 19 8 13 60 15 36 21 19 11 8 15 Nonparametric methods--------- 8 2 5 33 3 16 7 8 3 4 7 Other------------------------- 2 (2) 1 9 1 2 3 4 2 5 4 Miscellaneous: - Fluid dynamics---------------- 11 7 13 1 23 6 23 8 2 (2) 5 Electrodynamics--------------- 6 3 7 1 10 4 14 6 3 |------ 2 Elasticity theory.--------------- 5 3 8 1 11 2 14 4 1 (2) 2 Magneto-hydrodynamics-------- 2 1 2 (2) 2 (2) 6 1 1 ------ 1 Linear vibrations--------------- 9 6 10 1 20 3 18 7 2 ------ 4 Nonlinear vibrations------------ 5 4 7 1 10 2 11 5 1 ------ 2 Mathematical theory of com- puters and control devices----- 25 34 27 14 27 25 26 19 15 14 22 Programming of high-speed digi- tal computers---------------- 42 64 48 27 35 42 39 30 26 17 36 Linear programming------------ 22 26 22 23 13 46 15 18 7 17 20 Mathematical methods in opera- tions analysis----------------- 21 18 18 20 13 58 16 20 8 22 18 her------------------------- 6 4 6 3 6 5 7 11 8 4 8 No courses reported------------ 2 1 1 1 2 1 1 2 4 4 4 * Includes 53 respondents who did not specify the type of organizational unit in which they worked. * Less than 0.5 percent. 62 TABLE A–32.-Courses required for positions in mathematical employment for which formal training was hard to secure, 1960' Number Number Courses of times Courses of times cited cited Elementary courses: Topology: College algebra---------------------------- 16 Algebraic (combinatorial) topology- - - - - - - - - - 13 Trigonometry----------------------------- 9 Point set topology------------------------- 11 Analytic geometry------------------------- 12 Topological groups------------------------ 5 Differential and integral calculus-- - - - - - - - - - - 15 Homotopy theory------------------------- 1 Foundations: º Homology theory-------------------------- 1. Foundations of mathematics--- - - - - - - - - - - - - - 31 Lie groups-------------------------------- 2 Set theory-------------------------------- 31 Lie algebras------------------------------ 2 Mathematical logic------------------------ 102 Other------------------------------------------ Other------------------------------------ 14 || Geometry: Algebra and number theory: Advanced analytic geometry.-- - - - - - - - - - - - - - - 19 Higher algebra---------------------------- 41 Advanced Euclidian geometry----- - - - - - - - - - - 6 Theory of equations----------------------- 44 Non-Euclidian geometry- - - - - - - - - - - - - - - - - - - 6 Algebra of vectors and matrixes- - - - - - - - - - - - - 141 Projective geometry--------- - - - - - - - - - - - - - - 14 Theory of rings and fields- - - - - - - - - - - - - - - - - - 11 Algebraic geometry------------------------ 6 Abstract algebraic structures----- - - - - - - - - - - - 6 Differential geometry---------------------- 19 Tensor algebra (multilinear) - - - - - - - - - - - - - - - - 30 ther------------------------------------ 5 Theory of groups-------------------------- 16 || Probability and statistics: Lattice theory---------------------------- 19 Elementary statistics- - - - - - - - - - - - - - - - - - - - - - 73 Elementary number theory - - - - - - - - - - - - - - - - - 32 Mathematical statistics- - - - - - - - - - - - - - - - - - - - 145 Algebraic number theory.-- - - - - - - - - - - - - - - - - - 16 Statistical inference------------------------ 87 Analytic number theory-------------------- 21 Probability theory------------------------- 167 Other------------------------------------ 11 Information theory.------------------------ 103 Analysis: - Stochastic processes------------------------ 113 Differential equations----- - - - - - - - - - - - - - - - - - 27 Statistical decision theory - - - - - - - - - - - - - - - - - - 102 Advanced calculus------------------------- 38 Theory of games-------------------------- 121 Vector analysis---------------------------- 62 Correlation analysis------------------------ 135 Partial differential equations_ _ _ _ _ _ _ _ _ _ _----- 48 Sampling theory--------------------------- 126 Integral equations------------------------- 34 Design and analysis of experiments__ _ _ _ _ _ _ _ _ 168 Calculus of finite differences- - - - - - - - - - - - - - - - 122 Nonparametric methods-------------------- 92 Calculus of variations---------------------- 74 Other------------------------------------ 39 Series and Summability--------------------- 31 || Miscellaneous: Functional analysis------------------------ 17 Fluid dynamics--------------------------- 26 Linear spaces & operator theory_-_ _ _ _ _ _ _ _ _ _ _ 16 Electrodynamics-------------------------- 13 Hilbert Spaces----------------------------- 2 Elasticity theory.-------------------------- 24 Banach spaces---------------------------- 2 Magneto-hydrodynamics--- - - - - - - - - - - - - - - - - 10 Functions of real variables-------- - - - - - - - - - - 27 Linear vibrations-------------------------- 17 Functions of complex variables_-_ _ _ _ _ _ _ _ _ _ _ _ 38 Nonlinear vibrations----------------------- 33 Algebraic functions------------------------ 9 Mathematical theory of computers and con- Special functions-------------------------- 20 trol devices----------------------------- 205 Orthogonal functions----------------------- 40 Programming of high-speed digital computers- 242 Integral transforms------------------------ 35 Linear programming------------------------ 232 Harmonic analysis------------------------- 29 Mathematical methods in operations analysis- 190 Potential theory--------------------------- 21 Other------------------------------------ 62 Lebesgue measure and integration_ _ _ _ _ _ _ _ _ _ _ 12 Theory of conformal mapping--------------- 14 Numerical analysis------------------------ 223 Other------------------------------------ 16 1 Reasons most frequently given for difficulty in securing courses cited include: not offered at convenient locations, not in school curriculum, not offered at night or as extension course, not enough people interested to form class, and qualified teachers not available. 63 TABLE A-33–Course deficiencies noted by supervisors among persons recently considered for typical bachelor's level mathe- - matics positions under their supervision, 1960 Number Number Courses of times Courses of times - Cited cited Elementary courses: Topology: College algebra---------------------------- 40 Algebraic (combinatorial) topology---------- 4 Trigonometry----------------------------- 29 Point set topology------------------------- 3 Analytic geometry------------------------- 44 Topological groups------------------------ 2 Differential and integral calculus------------ 67 Homotopy theory-------------------------|------ Foundations: Homology theory--------------------------|------ Foundations of mathematics---------------- 50 Lie groups--------------------------------|------ Set theory-------------------------------- 37 Lie algebras------------------------------|------ Mathematical logic------------------------ 94 || Other------------------------------------------ Other------------------------------------ 8 || Geometry: Algebra and number theory: Advanced analytic geometry---------------- 27 Higher algebra---------------------------- 69 Advanced Euclidian geometry--------------- 2. Theory of equations----------------------- 54 Non-Euclidian geometry------------------- 2 Algebra of vectors and matrixes------------- 224 Projective geometry----------------------- 8 Theory of rings and fields------------------ 4 Algebraic geometry------------ ------------ 6. Abstract algebraic structures---------------- 2 Differential geometry---------------------- 7 Tensor algebra (multilinear)---------------- 6 Other------------------------------------ 1 Theory of groups-------------------------- 9 || Probability and statistics: Lattice theory---------------------------- 3 Elementary statistics---------------------- 196. Elementary number theory.----------------- 43 Mathematical statistics-------------------- 211 Algebraic number theory.------------------- 12 Statistical inference------------------------ 83 Analytic number theory.-------------------- 13 Probability theory------------------------- 256 Other------------------------------------ 11 Information theory.------------------------ 36 Analysis: Stochastic processes------------------------ 32 Differential equations---------------------- 87 Statistical decision theory.------------------ 47 Advanced calculus------------------------- 100 Theory of games-------------------------- 43 Vector analysis---------------------------- 103 Correlation analysis--------------- --------- 103 Partial differential equations---------------- 64 Sampling theory— — — — — - - - - - - - - - - - - - - - - - - - - - 94 Integral equations------------------------- 15 Design and analysis of experiments---------- 116 Calculus of finite differences---------------- 120 Nonparametric methods-------------------- 38 Calculus of variations---------------------- 27 Other------------------------------------ 33 Series and summability--------------------- 32 || Miscellaneous: Functional analysis------------------------ 7 Fluid dynamics--------------------------- 19 Linear spaces and operator theory.----------- 5 Electrodynamics-------------------------- 9 Hilbert spaces-----------------------------|------ Elasticity theory-------------------------- 10 Banach spaces----------------------------|------ Magneto-hydrodynamics-------------------|------ Functions of real variables------------------ 24 Linear vibrations-------------------------- 19 Functions of complex variables-------------- 49 Nonlinear vibrations----------------------- 8 Algebraic functions------------------------ 7 Mathematical theory of computers and control Special functions-------------------------- 8 devices--------------------------------- 113 Orthogonal functions----------------------- 18 Programming of high-speed digital computers- 207 Integral transforms------------------------ 24 Linear programming----------------------- 93 Harmonic analysis.------------------------ 18 Mathematical methods in operations analysis- 100 Potential theory--------------------------- 7 Other------------------------------------ 58 Lebesgue measure and integration----------- 2 Theory of conformal mapping--------------- 6 | Numerical analysis------------------------ 270 6 Other------------------------------------ 64 TABLE A–34.—Deficiencies cited by supervisors in the mathematics training of post-World War II mathematics graduates, by educational level and field of employment, 1960 Area of deficiency or remedy cited Number of times deficiency was cited For bachelor’s degree holders For doctor’s degree holders Private Non- Private Non- Total industry, Insur- || Govern-| profit Total industry, Insur- Govern-| profit excluding ance ment | Organi- excluding ance ment organi- insurance zations insurance zations Statistics------------------------------- 85 60 2 23 ------ 19 13 ------ 6 ------ Probability and statistics----------------- 61 34 6 21 l------ 25 | 16 ------ 9 ------ Mathematics and statistics--------------- 14 11 2 1 ------|------|-------|------|------|------ Actuarial science------------------------ 11 ------- 11 ------------|-------------------------|------ Applied statistical analysis---------------- 9 5 ------ 4 ------ 4 4 ------|------|------ Numerical analysis and application to - computers---------------------------- 60 47 ------ 12 1 ||-------------------|------|------ Numerical analysis---------------------- 48 30 1 17 ------ 12 8 ------ 4 ------ Training with computers----------------- 26 25 ------|------ 1 9 6 ------ 3 ------ Programming and data processing--------- 26 23 |------ 3 ------ 4 1 ------ 3 ------ Mathematical logic---------------------- 27 20 1 6 ------ 3 1 I 1 ------ Finite differences------------------------ 19 13 4 * !------|------|-------------|------------ Matrix algebra and theory of differences---- 15 9 ------ 6 ------ 1 1 ------|------|------ Computation techniques------------------ 14 4 ------ 10 ------|------|------------------------- Differential equations-------------------- 11 11 ------------------ 2 2 ------------|------ Number theory and analysis-------------- 9 9 ------|------------ 1 -------------|------ 1 Topology and algebra-------------------- 7 6 ------ 1 ------ 1 1 ------------------ Training in methods of approximation com- putations-----------------------------|------|-------|------|------|------ 3 I 1 1 ------ Applied mathematics-------------------- 100 70 3 25 2 16 12 l------ 4 ------ Broader training in the application of mathematics-------------------------- 23 4 18 ------ 1 33 27 1 5 ------ Greater ability in application of theory.----- 24 13 ------ 11 ------|------|-------------------|------ Practice in solving problems-------------- 12 11 1 ------|------|------------------------------- Application of mathematics to problems---- 9 9 ------|------------ 1 1 ------|------------ Orientation to applied Work--------------- 9 4 1 * ------|------|------------------------- Need more applied study, less theory.------|------|-------|------|------|------ 9 6 1 2 ------ Rnowledge of how to apply tools of trade-- 5 4 ------ ! ------|-------------------|------|------ Training in fundamentals----------------- 58 38 6 14 ------ 4 3 1 ------------ Basic mathematics----------------------- 16 13 2 ! ------|------|-------------------|------ Basic fundamentals---------------------- 14 9 2 1 2 1 1 ------------------ Basic understanding of course materials---- 12 8 1 3 ------|------|------------------------- Need more basic work, less theory.--------- 6 6 ------------------|------|------------------------- Higher standards of education (college) - - - - 54 8 18 28 l------ 2 2 ------------------ Insufficient mathematics----------------- 50 23 5 21 1 ||------------------------------- Mathematics in high School--------------- 37 21 7 9 ------|------|------------------------- Better teachers-------------------------- 13 9 ------ 4 ------ 6 6 ------------------ Too much specialization------------------ 2 -------|------ 1 1 16 13 ------ 2 1. Rigorous training------------------------ 14 10 2 1 1 1. 1 ------------------ More emphasis on mathematics major- - - - - 8 5 ------ 3 ------ 6 3 ------ 3 ------ Abstract mathematics and mathematical theory------------------------------- 7 2 ------ 5 ------|------------------------------- Mathematical maturity------------------ 7 6 |------ ! ------|-------------|------------------ Integrated course of study---------------- 5 ------------- 5 ------ 2 2 ------------------ Broad background----------------------- 4 3 1 ------------ 3 3 ------------------ Introduce more courses in colleges--------- 7 7 ------------------|-------------------------|------ Higher standard of performances---------- 6 6 ------------|------|-------------|------------------ Underlying principles lacking------------- 5 5 ------------|------|------|-------------|------------ Need more emphasis on training, less re- search-------------------------------- 4 3 ------|------ 1 1 ------------------- 1 Mathematical theory.-------------------- 3 3 ------------|------ 2 2 ------------------ Integration of pure and applied mathematics-------|-------|------|------|------ 4 3 ------ 1 ------ Modern courses; obsolete methods now taught-------------------------------|------|-------|------|------|------ 4 ------------------ 65 TABLE A–35.—Deficiencies cited by supervisors in scientific or engineering training,” other than mathematics, of post-World War II mathematics graduates, by educational level and field of employment, 1960 Area of deficiency or remedy cited Number of times deficiency was cited For bachelor's degree holders For doctor's degree holders Private Non- Private Non- Total industry, Insur- || Govern- profit Total industry, Insur- || Govern-| profit excluding ance ment organi- excluding ance ment | Organi- insurance zations insurance zations Physical sciences: Physics---------------------------- 225 153 |------ 70 2 33 27 ------ 5 1 Applied physics--------------------- 5 4 ------------ 1 3 3 ------|------------ Physics fundamentals---------------- 4 3 |------ ! ------ 4 4 ------------|------ Theoretical physics------------------ 1 1 ------------------ 5 2 ------ 3 ------ Engineering physics------------------ 15 15 ------|------|------|------|-------|------|------|------ Physics, chemistry and engineering---- 3 3 ------|------|------ 2 -------|------ 2 ------ Physics and chemistry--------------- 18 18 ------|------------|---- --------------------------- Chemistry-------------------------- 1 1 ------|------------ 3 3 ------|------|------ Engineering: Engineering------------------------- 113 71 2 40 |------ 8 7 ------|------ 1 Electronics, mechanics, etc.- - - - ------- 25 18 ------ 7 ------|-------------------------|------ Engineering Orientation-------------- 16 15 |------ 1 ------ 9 9 ------|------------ Engineering fundamentals------------|------|-------|------|------|------ 3 3 ------------|------ Electrical engineering---------------- 11 11 ------|------|------ 3 3 ------|------|------ Fluid dynamics, aerodynamics, etc.----- 5 5 ------|------|------ 3 -------|------ 3 ------ Communications: Technical writing-------------------- 56 37 3 16 |------ 5 5 ------------|------ English and Speaking ability---------- 50 26 7 13 4 6 3 ------ 2 1 Communication of ideas-------------- 16 10 |------ 6 ------ 3 3 ------------|------ Application of mathematics--------------- 65 53 3 9 |------ 21 17 ------ 4 ------ Application of theory-------------------- 32 20 ------ 12 ------ 5 2 ------ 3 ------ Inadequate knowledge of applied fields----- 6 5 ! ------------ 17 13 ------ 4 ------ Too much specialization------------------ 7 6 ------ 1 ------ 11 9 |------ 2 ------ Integration among courses---------------- 11 11 ------|------|------ 3 2 ------ 1 ------ Too general education-------------------- 8 4 1 3 ------ 2 2 ------------|------ On the job training---------------------- 7 4 ------ 3 ------|-------------|------|------|------ Experience----------------------------- 4 3 ------ 1 ------ 3 3 ------------------ Analysis work in general----------------- 7 5 ------ 2 ------|------|-------|------------|------ Logic---------------------------------- 2 ------------- 2 ------ 5 5 ------------------ Broad background----------------------- 4 2 ------ 2 ------ 2 2 ------|------------ Fundamentals--------------------------- 11 8 |------ 3 ------|-------------|------------|------ Rigorous training------------------------ 5 5 ------------------|------|------------------------- Natural Sciences------------------------- 5 4 ------ 1 ------|------------------------------- Operations research---------------------- 1 1 ------------|------ 3 3 ------|------------ Liberal arts----------------------------- 13 5 3 5 ------|------|-------|------|------|------ Humanities----------------------------- 11 8 1 1 1 ||-------------|------|------------ Foreign languages----------------------- 6 4 ------ 2 ------ 5 2 ------ 3 ------ Economics------------------------------ 8 4 ------ 2 2 ||------|------------------------- Psychology----------------------------- 6 5 ------ 1 ------|-------------------|------|------ Business aptitude------------------------ 4 4 ------|------------ 4 3 ------ 1 ------ 1 Although supervisors were asked about training deficiencies in science and engineering only, information voluntarily provided on deficiencies in other areas has been included in the table. &6 Appendix B SCOPE AND METHOD Scope of Survey This survey was concerned with mathematical employment outside of teaching. Therefore, information was solicited from individuals em- ployed primarily in mathematical work in private industry, the Federal Government, and private nonprofit organizations but not from personnel in universities or colleges. Since the intention was to include all professional personnel engaged primarily in mathematical work whether or not they had the job title of mathematician, the criteria for inclusion were in terms of the duties of the position held rather than the occupational title. An individual was considered within the scope of the survey if he occupied a position meeting all three of the following criteria: 1. It is a full-time position. 2. Its performance requires knowledge equal at least to that provided by a 4-year college course with a major in mathematics. 3. Mathematics predominates over all other fields in either (a) the nature of the work done in the position or (b) educational prerequisites for the position. In order to locate individuals in positions meeting the above criteria, a list of employers was obtained by subsampling a sample of companies previously drawn for a survey of scientific and technical personnel in industry in 1959. The latter was a sample of companies, stratified by industry and size, drawn chiefly from the universe of employers covered by the Federal Old-Age and Survivors Insurance program in 1956, and includ- ing about 10,500 companies representing all non- agricultural industries except those known to employ few scientific and technical personnel." For the 1960 survey of mathematical employment, all companies reporting that they had mathe- 1 For further information on the sample from which the subsample was drawn, see Appendix B, Scope and Method, in Scientific and Technical Personnel in American Industry, Report on a 1959 Survey, National Science Foundation NSF 60–62, 1960. officials of the National Science Foundation. maticians on their payroll in 1959 were included (except for companies in some small-size classes which reported only one or two. In addition, companies which had reported scientists without indicating their specialty, had reported no mathe- maticians, or had not submitted any report at all were subsampled. All Federal Government agencies, except certain national security agencies, and all private nonprofit organizations believed to employ mathematicians were also included. Employer questionnaires were sent to 1,778 companies in private industry, to 37 Federal Government agencies, and to 24 nonprofit organi- zations. Replies were received from 1,486 com- panies (an 84-percent response), and from all Government agencies and nonprofit organizations queried. All together, these employers reported distribution of detailed questionnaires to 14,367 persons in positions believed to meet the criteria stated above. Individuals returned 10,737 or about 75 percent of the detailed questionnaires. Upon examination, 755 of these, largely from a single Government agency, were determined to be outside the scope of the survey, leaving 9,982 replies which were analyzed in the main body of the report. Questionnaire Content Two reporting forms were used for this survey (appendix C). Schedule A requested employers to furnish data on total employment and on the number of filled and vacant positions meeting the criteria described above. The form also asked employers to distribute questionnaires to employ- ees in mathematical positions. Schedule B, the employee questionnaire, requested detailed data on the characteristics of mathematical positions and of the persons in such positions. The basic content for the questionnaires was suggested by the Mathematical Association of America's study committee for this project and The questionnaires were prepared by the Bureau of 67 Labor Statistics, with advice and assistance from the Association. The U.S. Bureau of the Budget, which approves Federal Government requests for reports from private industry, convened a panel of industry representatives to review the report forms, and this review resulted in many sugges- tions which were incorporated in the question- naires. The tabulation plans and this report were prepared in the Bureau of Labor Statistics with the advice of members of the Mathematical Association of America and officials of the National Science Foundation. Data Collection Two approaches were used in obtaining the cooperation of employers included in the survey. The Government agencies and about 60 companies believed to be employers of the largest numbers of persons in mathematical work were visited by representatives of the Bureau of Labor Statistics, who explained the purpose of the survey, discussed procedures for distributing the employee question- naires, and arranged for delivery of the necessary questionnaires to the company offices. All other employers were mailed an explanatory letter and a supply of questionnaires believed to be sufficient to permit distribution to each of their employees engaged in mathematical work. The covering letter and followup letters to employers who did not respond, or whose employee responses were low, are reproduced in appendix C. Completed questionnaires were returned during the spring and summer of 1960. A large majority of the companies replied to the survey by returning completed questionnaires without adding any qualifying comments. How- ever, officials of a few large companies contacted by mail, and others among those contacted in person, reported difficulty in singling out em- ployees who met the position criteria for inclusion in the survey. Some comparies had information on the skills or educational background of each employee and sent questionnaires to all who had a mathematics background, regardless of current work. Others left distribution to supervisors in the divisions most likely to employ persons with professional mathematical training. It was suggested to Government agencies that they distribute questionnaires to employees classi- fied as mathematicians, mathematical statisticians, and actuaries. One agency thought that some of its employees classified as physicists met the cri- teria. The Department of the Army believed it had employees primarily engaged in mathematical work among its analytical statisticians and its management analysts and distributed question- naires to such employees above the trainee level (grades GS-11 and above), as well as to employees in the three recommended classifications. Tabulation of Replies From Individuals To eliminate replies from persons in positions which required little or no mathematics, returned questionnaires were excluded from the tabula- tions unless a respondent indicated one of the following: The minimum mathematical education re- quired for his work was equal to or above a bachelor's degree with a major in mathe- matics He had the job title of mathematician or re- quired calculus or advanced mathematical courses to perform his work or had a college degree with a major in mathematics These criteria resulted in the elimination from the study of about 3 percent of the individual replies, excluding those received from the Department of the Army. About one-third of the replies from Army employees were eliminated by the added criteria. - Within the scope of the survey, as established by the two sets of criteria, were a number of persons who reported that the minimum mathe- matical education required for their positions was less than the equivalent of a bachelor's degree with a major in mathematics, but whose employers believed they met the criteria for inclusion. Two objectives were sought by the inclusion of these individuals. One was to secure information from persons who, although primarily engaged in mathematical work, were operating at the lower professional levels. The second was to include those individuals, recently out of school, who might believe they were not working at a pro- fessional level but whose employers considered them as professionals, although possibly in a trainee status. In addition, no doubt, among the respondents reporting that they did not need a college major in mathematics in their positions were some working primarily as engineers or physicists. The method of contacting individuals (which precluded control over nonrespondents) did not provide an adequate basis for a probability sample 68 TABLE B–1.—Percent distribution of respondents in private findustry to survey of mathematical employment, 1960, and mathematicians, 1959 Persons in mathemati- Mathe- Industry cal employ-| maticians, ment 1960 1 || 1959 3 Total: - Number----------------------- 7, 171 11, 300 Percent----------------------- 100. 0 100. 0 Aircraft and parts------------------ 27, 7 29. 6 Electrical equipment---------------- 17. 2 16. 6 Machinery, except electrical---------- 8. 4 8. 3 Professional and scientific instruments- 2. 6 1. 7 Other durable goods manufacturing--- 10. 4 11. 8 Petroleum products and extraction---- 6. 3 4. 5 Chemicals------------------------- 4. 4 4. 9 Other nondurable manufacturing----- 2.4 3. 1 Insurance-------------------------- 12. 9 } 18. 5 Service industries, except insurance--- 7. 3 - Other nonmanufacturing------------ ... 4 1. 0 1 Excludes companies with fewer than 100 employees in all occupations, About 1,000 persons were engaged in mathematical employment in these small-size companies in 1960. 3 Source: Scientific and Technical Personnel in American Industry, Report on a 1959 Survey, NSF 60–62, (p. 28). TABLE B–2.—Percent distribution of respondents in private tndustry to survey of mathematical employment, 1960, and nathematicians, 1959, by size of employing company Persons in mathe- Mathe- Size of employing company (number of employees) matical maticians, employ- 1959 2 ment, 19601 Total: Number----------------------- 7, 171 11, 300 Percent----------------------- 100. 0 100. 0 25,000 or more employees------------ 52.4 } 73. 5 5,000–24,999 employees— — — — — — — — — — — — — 28. 0 e 1,000-4,999 employees-------------- 15. 9 11. 7 500–999 employees----------------- 2. 5 4, 9 100-499 employees------------------ 1. 2 4, 8 Under 100 employees---------------|-------- 5. 1 1 See footnote 1 table B-1. 2 Source: Mathematicians—Scientific and Techniac! Personnel in American ſººn Report on a 1959 Survey, National Science Foundation, NSF 60–62 p. 31). which would permit making detailed estimates of the total number of persons in mathematical employment with specific chracteristics. Never- theless, it appears that the replies from employees working in private industry are reasonably repre- sentative by industry, although small-size com- panies are underrepresented. This is shown in tables B–1 and B-2, which compare the percentage distributions of the replies to this survey and of estimated total employment of mathematicians in private industry in 1959. Estimates of Total Employment in Mathematical Work Approximately 22,000 persons were estimated to be employed in mathematical work other than teaching in mid-1960. Of these, more than 17,000 were employed in private industry in companies with a total employment (all occupations) of 100 or more, estimated on the basis of company reports (schedule A, appendix C) of the number who met the criteria for inclusion in the survey, adjusted for Sampling ratios and company non- response. (See appendix table B-3.) Almost 3,000 were in the Federal Government and 300 in nonprofit organizations. Information from other sources indicates that about 1,000 additional persons were employed in companies with fewer than 100 employees, and another 500 in State and local governments. TABLE B-3.−Estimates of total employment in mathematical work other than teaching, 1960 survey Private Federal Nonprofit Basis for estimate Total industry 1 || Govern- organiza- Iment 2 tions Total.------ 20, 300 17, 100 2,900 300 Employer reports adjusted for Sam- pling ratio------ 17, 500 14, 300 2,900 300 Estimate for non- TeSp0InSe-- - - - - - - 2,800 | 2,800 --------|-------- 1 See footnote 1 to table B-1. º 2 Excludes certain national Security agencies. 69 Appendix C - QUESTIONNAIRES AND COWERING LETTERS B.L.S. No. 2651 a A Survey of F- MATHEMATICAL EMPLOYMENT 0THER THAN TEACHING Conducted for the NATIONAL SCIENCE FOUNDATION By the U.S. DEPARTMENT OF LABOR Bureau of Labor Statistics L Your reply will be held in STRICT CONFIDENCE SCHEDULE A QUESTIONNAIRE FOR COMPANIES AND AGENCIES Budget Bureau No. 44–5928. Approval expires September 80, 1960. -1 Change address if incorrect PURPOSE OF THE SURVEY This survey is being conducted, under the sponsorship of the Mathematical Association of America as well as the National Science Foundation, to find out what kinds of work mathema- ticians do in industry and government, what training they have had and need, and other important facts about their employ- ment. The major purpose is to provide information needed by the mathematics profession in planning improvements in mathe- matics curriculums, to meet employers’ rapidly changing and growing requirements. The survey findings will also be used by the National Science Foundation in developing policies to strengthen the country's scarce personnel resources in this critically important field. Your reply will be held in STRICT CONFIDENCE. It will be seen only by sworn employees of the Bureau of Labor Statistics and the National Science Foundation, and no data will be released which would identify any company, agency, or individual. PLAN OF THE SURVEY The survey covers a carefully selected Sample of companies and of government and nonprofit agencies, and utilizes two different questionnaires—Schedule A, which you are now reading, and Schedule B, several copies of which are enclosed. You are requested to: 1. Complete Schedule A and mail it directly to the Commissioner of Labor Statistics (see instructions below). A return envelope is enclosed. 2. Distribute Schedule B to all professional personnel engaged in primarily mathematical work or work requiring primarily mathematical training (see instructions on the reverse side). Every such person should be asked to fill out Schedule B as soon as possible and mail it directly to the Commissioner of Labor Statistics. A return envelope is enclosed for each Schedule B. INSTRUCTIONS FOR COMPLETING SCHEDULE A (QUESTIONNAIRE FOR COMPANIES AND AGENCIES) We need a reply to Schedule A from your company or agency even if it has no professional personnel engaged in primarily mathematical work or work requiring primarily mathematical training. Please write “None” where appropriate rather than leave a blank space. If exact figures are not available, reason- able estimates will be satisfactory. Coverage of the schedule. Please supply information for your company only, excluding data on separately incorporated parent, subsidiary, or affiliated companies. If this is not feasi- ble and a consolidated return is sent in, please list in Item 8 on the reverse side all subsidiaries and affiliates covered by the data. If extra copies of Schedule A would be helpful, they may be obtained on request. Mail completed Schedule A to: COMMISSIONER OF LABOR STATISTICS, U.S. DEPARTMENT OF LABOR, Washington 25, D.C. Name and title of official submitting reply If you would like a copy of the release on the survey findings, please check D (Please print or type) 16-73580-1 72 INSTRUCTIONS FOR DISTRIBUTING SCHEDULE B (QUESTIONNAIRE FOR INDIVIDUALS) Please distribute Schedule B to all professional personnel whose position meets all three of the following criteria: 1. It is a full-time position; 2. Its performance requires knowledge equal at least to that provided by a 4-year college course with a major in mathematics; and 3. Mathematics predominates over all other fields in either (a) nature of the work done in the position, or (b) educational prerequisites for the position. In many organizations these criteria may be met not only by positions with the title of mathematician but also by some with other job titles. Operations research, programming for computers, quality control, mathematical statistics, and actuarial work are among the areas likely to include some personnel who should receive Schedule B. If you need more copies of Schedule B, please indicate number of additional copies needed. . . . . . * * * * * * * * * * 1. How many full-time positions are there in your organization, whether now filled or vacant, which meet the above criteria? . . . . . . . . . . . . . . . . . . . . . * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * A. Of these, how many are now filled?..... e & e e s e g º e tº º e º tº ſº e º ºs º e º e º e º e º ºs º e º s is e º e º e s a s e e s s e º a s sº e º ºs e º 'º a s (This number should equal the number of persons asked to fill out Schedule B.) B. How many are now vacant?... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . & it s tº e º a tº e º e º ºs e s tº e º e o e & s a tº e e s e a e º e s is a s e e 2. Please give total number of persons, in all occupations and activities, employed in your organization in the United States as of January 1960. (Include part-time as well as full-time employment.). . . . . . . . . . . . . . . .t 3. Does this reply cover any separately incorporated parent, subsidiary, or affiliated companies? Yes D. No DT. If “Yes,” please list their names: 4. Is your company a separately incorporated subsidiary of any other company ? Yes D. No [] If “Yes,” give name of parent company ----------------------------. 16-75530-1 U.S, GOVººDººl'ſ Pººl ºf NG OFFICE Mail completed Schedule A to: COMMISSIONER OF LABOR STATISTICS U.S. DEPARTMENT OF LABOR Washington 25, D.C. 73 A Survey of MATHEMATICAL EMPLOYMENT OTHER THAN TEACHING Conducted for the NATIONAL SCIENCE FOUNDATION By the U.S. DEPARTMENT OF LABOR Employer Bureau of Labor Statistics Code Number Dear Sir: The increasing importance of science to our country’s security and welfare makes it imperative to have current information on the Nation’s resources of scientific personnel. Since the rapidly growing field of mathematics is one on which information is now especially needed, the National Science Foundation and the Mathematical Association of America are sponsoring a survey of mathematical employment outside of teaching. The findings will help the mathematics profession in planning improvements in mathematics curriculums, to meet rapidly changing and grow- ing requirements. They will also be used by the Foundation in developing policies for strengthening the country's scientific manpower resources. In addition, the findings will be valuable to the profession in recommending policies for recruitment and utilization of mathematicians. Since you and your employer are part of a carefully designed sample, a response from you is very important to the success of the survey. You are not asked to sign the questionnaire. Furthermore, your reply will be held in strict confidence and will be seen by no one except sworn employees of the Bureau of Labor Statistics and the National Science Foundation. No data will be released which would identify any individual, company, or agency. Please fill out this questionnaire as soon as possible and return it, in the envelope provided, directly to: COMMISSIONER OF LABOR STATISTICs U.S. Department of Labor Washington 25, D.C. We shall be very grateful for your prompt cooperation. Very truly yours, &ow EwAN CLAGUE, Commissioner of Labor Statistics. 74 our rep €HEDULE & Budget Bureau No. 44–5929. Y ly will be held in S Å. expires September 30, 1960. $ºl ºf CONFIºENCE QUESTION:NAHää FOR INDIVIDUALS B.L.S. No. 265ib 1. Date of birth: 2. Sex: Male D Female D BºſcàTIONAL BACKGROUND 3. List below all college and graduate training, including education completed after receipt of your highest degree. NAME OF INSTITUTION AND LOCATION MAJOR SUBJECT DEGREE * º *If not applicable, enter instead the number of semester hours completed. PROFESSIONAL EXPERIENCE 4. How many years have you had of full-time professional experience of all kinds and, of these, how many were spent with each of the types of employers shown below? (Exclude assistantships and other part-time employment while attending school.) NUMBER OF YEARS A. Private industry (include facilities operated by private industry under government contracts). . . . . . . . . . . . . . . . . . . . - B. Government: international, Federal, State, or local (exclude universities and colleges). . . . . . . . . . . . . . . . . . . . . . . . . . C. University or college (include facilities operated by universities under government contracts). . . . . . . . . . . . . . . . . . . . D. Other nonprofit organizations (include facilities operated by nonprofit organizations under government contracts). . . E. Self-employment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. By how many different companies, government bureaus, educational institutions, or other employers have you been MAJOR CURRENT POSITION 6. Job title 7. Government grade (if applicable) 8. A. Check the one category which best describes the smallest organizational unit to which you belong in your major current position. [] (0) Computing laboratory [] (5) Research laboratory or other research unit [] (1) Mathematics unit [T] (6) General technical staff D (2) Statistical unit […] (7) Production unit D (3) Engineering unit D (8) Administration unit (e.g., budget office) DJ (4) Operations research unit [T] (9) Cther (Speciſ) 75 B. Check the one statement which best describes your operational status in your major current position. D (0) Administrator (decision making, little technical supervision) [] (1) Supervisor (mostly technical supervision) Mathematical practitioner who: [ ] (2) works primarily with other mathematicians [] (3) works individually, with little or no consultation with others [I] (4) works as an individual consultant to others [...] (5) works as a member of a team made up chiefly of non-mathematicians [...] (6) Other (Speciſy) 9. This item deals with the mathematical work you do in your major current position. A. In the box next to the function in which you are chiefly engaged in your mathematical work, enter “1.” If pertinent, designate second and third most important functions with “2” and “3.” [T] (0) Basic research in the natural sciences and engi- [T] (4) Technical services allied to sales, promotion, or distribution neering (1) Applied research and development in the natural D (5) Teaching and training sciences and engineering [ ] (2) Nontechnological research, including marketing and [...] (6) Administration other economic research [ ] (3) Technical services allied to production [T] (7) Other (Specify) B. In the box next to the subject-matter area from which chiefly come the problems you deal with in your mathematical work, enter “l.” If pertinent, designate second and third most important areas with “2” and “3.” [ ] (0) Basic mathematics [ ] (5) Social sciences including economics, and psychology [ ] (1) Phyde. chemistry [T] (6) Insurance [ ] (2) Earth sciences, meteorology [I] (7) Office and business administration [T] (3) Engineering [T] (8) Production and inventory [ ] (4) Biological, medical, and agricultural sciences [] (9) Other (Specify) C. In the box next to the mathematical field you use most in solving problems in your mathematical work, enter “1.” If pertinent, designate second and third most important fields with “2” and “3.” D (0) Computer techniques [T] (5) Actuarial mathematics [T] (1) Numerical analysis, theory of computation [T] (6) Logic, theory of sets [I] (2) Analytical mechanics including fluid dynamics, elec- [] (7) Algebra, including theory of numbers tromagnetics (3) Analysis [I] (8) Topology and geometry [T] (4) Probability and statistics [T] (9) Cther (Specify) 10. A. Indicate by a check your estimate of the minimum mathematical education needed to perform the duties of your major current position. Please base your answer on the nature of your position rather than your personal qualifications. [ ] (0) Doctor's degree in mathematics [...] (2) Bachelor's degree in mathematics & [ ] (1) Master's degree in mathematics [I] (3) Less than a college major in mathematics 76 B. Check the phrase which best describes the extent to which you normally use in your major current position the amount of mathe- matical education you estimated for it in Item 10 A. [] (0) Almost always [T] (2) Less than half the time [T] (1) At least half the time [] (3) Almost never C. Printed on pp. 4-5 are titles of typical courses in undergraduate and graduate mathematics. Check in the appropriate boxes in the left-hand columns of the list those courses (including prerequisite courses) which, in your opinion, comprise the minimum requirements in mathematics for performing the duties of your major current position. Spaces are provided if you wish to specify courses not on the list. D. During the past 10 years have you tried and found it difficult to get formal training in any of the subjects you checked in answering Item 10 CP Yes D No [I] If “Yes,” indicate such courses with a second check in the appropriate boxes in the left-hand columns of the list on pp. 4-5. E. If you have double checked any courses in answering Item 10 D, please comment on the reasons for your difficulty in obtaining formal F. In your opinion, does the performance of your major current position require academic training in any scientific or engineering fields other than mathematics? Yes [] No D If “Yes,” please specify these fields. CURRENT PROFESSIONAL INCOME 11. This question asks about your professional income and is optional. A. Please check your annual income from your major current position. (11) D Below $4,000 - (23) [T] $7,000– 7,999 - (32) [T] $11,000–11,999 (12) D $4,000-4,999 (24) [] $8,000– 8,999 (33) [T] $12,000–14,999 (21) D $5,000–5,999 (25) [] $9,000– 9,999 (41) [I] $15,000–19,999 (22) D $6,000–6,999 (31) [] $10,000–10,999 (51) D $20,000 and over B. Please check additional professional income you earned during past year. (Exclude nonprofessional income, such as from investments etc.) (0) D None (3) D $1,000–1,499 (6) [T] $3,000–3,999 (1) D $1 –499 (4) D $1,500–1,999 (7) [] $4,000–4,999 (2) [] $500-999 (5) [I] $2,000–2,999 (8) [I] $5,000 and over C. If you indicated any additional professional income in Item 11 B, please check the source or sources of such income. (0) College or university teaching (3) Writing D (1) High school teaching D (4) Consultation D (2) Other teaching {Speciff)- D (5) Other nonteaching work (Specify) IIEMS FOR PERSONS WITH SUPERVISORY DUTIES (If your major current position involves no supervisory duties, check here [] and omit Items 12–14.) 12. Please give the total number of employees, in all kinds of work, under your direct or indirect supervision. . . . . . . . . . . 13. Of this number, how many are professional personnel engaged in primarily mathematical work or work requiring pri- marily mathematical training, the performance of which requires knowledge equal at least to that provided by a 4-year college course with a major in mathematics?... . . . . . . . . . . . & a tº e g tº s º g tº gº tº g is e g g ſº tº e s º w a * * is 4 & 2 × 4 + æ & e º 'º s as º is tº gº 77 14. If you reported any employees in Item 13, please answer Item 14; otherwise omit it. A. Check in the appropriate boxes in the right-hand columns of the list on this and the next page those courses (including prerequisite courses) which, in your opinion, comprise the minimum requirements in mathematics for a typical mathematics position at the bachelor's level under your supervision. Spaces are provided if you wish to specify courses not on the list. B. Among persons recently considered for such positions, have you noted deficiencies in meeting the minimum course requirements you checked in answering Item 14 A2 Yes [T] No [] If “Yes,” indicate with a second check in the appropriate boxes in the right- hand columns of the list on this and the next page the courses in which you have found such deficiencies. C. Have you noted deficiencies in the mathematics training of post-World War II mathematics graduates? Yes [T] No D If “Yes,” please comment on the ways this training might be improved. (1) Training for bachelor's degree (2) Training for Ph.D. degree D. Have you noted among post-World War II mathematics graduates educational deficiencies in scientific or engineering fields other than mathematics? Yes D No [T] If “Yes,” please comment on the additional types of training that would be helpful to your organi- zation. (1) Training for bachelor's degree (2) Training for Ph.D. degree UNDERGRADUATE AND GRADUATE MATHEMATICS COURSES Answers to Answers to Answers to Answers to I O C and D (p. 3) - 14 A and B (p. 4) 10 C and D (p. 3) 14 A and B (p. 4) Elementary courses Algebra and number theory [] (01) College algebra. . . . . . . . . . . . . . . . . . . . . . . . . . . . . - - - [] [...] (11) Higher algebra. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D [ ] (12) Theory of equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . D [] (02) Trigonometry. . . . . . . * * * * * * * * * - e < * * ... • * * * * * * * * * * * * [T] - e [] (13) Algebra of vectors and matrixes . . . . . . . . . . . . . . . . . . [] D (03) Analytic geometry. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . [T] [...] (14) Theory of rings and fields. . . . . . . . . . . . . . . . . . . . . . . D D (15) Abstract algebraic structures. . . . . . . . . . . . . . . . . . . . . . [...] [T] (04) Differential and integral calculus. . . . . . . . . . . . . . . . . . [T] & 4 : [] (16) Tensor algebra (multilinear). . . . . . . . . . . . . . . . . . . . . D. Foundations [T] (17) Theory of groups. . . . . . . . . . . . . . … • . . . . . . . . . . . . . . . D [I] (05) Foundations of mathematics. . . . . . . . . . . . . . . . . . . . . . [I] D (18) Lattice theory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D 19) Elementary number theory. . . . . . . . . . . . . . . . . . . . * * * [] [] (06) Set theory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . [T] [T] ( ) mentary ry [T] (20) Algebraic number theory. . . . . . . . . . . . . . . . . . . . . . . . [] D (07) Mathematical logic. . . . . . . . . . . . . . . . . . . . . . . . . . . . . [I] [] (21) Analytic number theory. . . . . . . . . . . . . . . . . . . . . . . . . D D (08) Other (Specift) D (22) Other (Specifi) D 78 UNDERGRADUATE AND GRADUATE MATHEMATICS COURSE5–Continued Answers to - Answers to Answers to Answers to 10 C and D (p. 3) 14 A and B (p. 4) 10 C cºnd D (p. 3) 14 A and B (p. 4) Analysis Geometry [T] (25) Differential equations. . . . . . . . . . . . . . . . . . . . . . . . - - - [T] [] (61) Advanced analytic geometry. . . . . . . . . . . . . . . . . . . [T] [T] (26) Advanced calculus. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . [] [T] (62) Advanced Euclidian geometry. . . . . . . . . . . . . . . . . . . . [ ] D (27) vector analysis........ . . . . . . . . . . . . . . . . . . . . . . . . [] D (63) Non-Euclidian geometry. . . . . . . . . . . . . . . . . . . . . . . . . [] - 64) Projecti try . . . . . . . . . . . . . . . . . . . . . . . . . . . . . [] (28) Partial differential equations. . . . . . . . . . . . . . . . . . . . . . [T] [...] (64) Projective geometry [T] D - [] [T] (65) Algebraic geometry. . . . . . . . . . . . . . . . . . . . . . . . . . . . . [] (29) Integral equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . - [] (66) Differential geometry. . . . . . . . . . . . . . . . . . . . . . . . . . . . [] [] (30) Calculus of finite differences. . . . . . . . . . . . . . . . . . . . . [T] [] [] (67) Other (Specift) [T] (31) Calculus of variations. . . . . . . . . . . . . . . . . . . . . . . . . . . [T] [] (32) Series and summability. . . . . . . . . . . . . . . . . . . . . . . . . . [] Probabili d statisti D (33) Functional analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . [] Q ty and statistics [ ] (70) Elementary statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . [] D (34) Linear spaces & operator theory (incl. spectral theory). [T] [ ] [ ] --- (71) Mathematical statistics. . . . . . . . . . . . . . . . . . . . . . . . . . . [] (35) Hilbert spaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . [T] [I] D (72) Statistical inference. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . [] (36) Banach spaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . [T] ) p [T] (73) Probability theory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . [I] [I] (37) Functions of real variables. . . . . . . . . . . . . . . . . . . . . . . [T] [ ] (74) Inf - h [T] nformation theory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . [T] (38) Functions of complex variables. . . . . . . . . . . . . . . . . . . [T] [ ] (75) Stochasti [] - tochastic processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . [] (39) Algebraic functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . [ ] [ ] (76) Statistical decision theory. . . . . . . . . . . . . . . . . . . . . . . . [T] D (40) Special functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . [T] [I] (77) Th f *- [T] eory of games . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . [] (41) Orthogonal functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . [T] [ ] (78) Correlation analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . [T] [] (42) Integral transforms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . [I] [] (79) S ling th [ ] ampling theory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D (43) Harmonic analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . [T] - [ ] (80) Design and analysis of experiments. . . . . . . . . . . . . . . . [] D (44) Potential theory........ . . . . . . . . . . . . . . . . . . . . . . . . . [ ] [ ] (81) Nonparametric methods. . . . . . . . . . . . . . . . . . . . . . . . . [] [T] (45) Lebesgue measure and integration. . . . . . . . . . . . . . . . . [] & [ ] (82) Other (Speciſ) D [I] (46) Theory of conformal mapping . . . . . . . . . . . . . . . . . . . . [I] D (47) Numerical analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . [ ] Miscell D (48) Other (Specifi) [T] $CGº || C Liº OUS [ ] (85) Fluid dynamics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . [] [] (86) Electrodynamics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . [I] Topology [] (87) Elasticity theory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . [] [] (51) Algebraic (combinatorial) topology . . . . . . . . . . . . . . . . [] [º] (88) Magneto-hydrodynamics . . . . . . . . . . . . . . . . . . . . . . . . . [] [I] (52) Point set topology. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . [T] [T] (89) Linear vibrations. . . . . . . . . . . . . . . . . . . . . . . . . . . . - * * * D [] (53) Topological groups. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . [T] D (90) Nonlinear vibrations. . . . . . . . . . . . . . . . . . . . . . . . . . . . [] [] (54) Homotopy theory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . [ ] [ ] (91) Mathematical theory of computers and control devices. D [I] (55) Homology theory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D [ ] (92) Programming of high-speed digital computers. . . . . . [ ] [I] (56) Lie groups. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . [] [...] (93) Linear programming. . . . . . . . . . . . . . . . . . . . . . . . . . . . [I] D (57) Lie algebras. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . [T] [] (94) Mathematical methods in operations analysis. . . . . . . [T] [] (58) Other (Specift) [T] D (95) Other (Specift) [T] Mail completed schedule to: COMMISSIONER OF LABOR STATISTICS U.S. Department of Labor Washington 25, D.C. 79 U.S. DEPARTMENT OF LABOR BUREAU OF LABOR STATISTICs WASHINGTON 25, D.C. Dear Sir: The expanding demand for mathematicians and the radically changing nature of mathematical work has created an urgent need to re-assess mathe- matics education. To help obtain the information required for planning improvements in mathematics curricula, the National Science Foundation has provided a grant to the Mathematical Association of America to make a study of this profession. To provide the data required for this study, the Bureau of Labor Statistics has been asked to conduct a survey of mathematical employment other than teaching. The survey will obtain detailed information on the educational demands of different types of mathematical positions and the background and qualifications of the mathematicians currently employed. We believe the results of this survey will not only contribute to the objectives of the National Science Foundation but will also be of great and direct benefit to employers of mathematicians. Your participation in this survey is extremely important to its success, whether your organization employs mathematicians or not. Like all the companies and agencies to which we are mailing questionnaires, your organization is an essential part of a carefully designed sample. The survey utilizes two questionnaires, Schedule A and Schedule B, copies of which are enclosed. Please read schedule A first, since it contains a statement of the purposes and plan of the study and instruct- tions for handling the two questionnaires. All replies will be kept strictly-confidential. They will not be published in any way which would permit the identification of any company, agency, or individual. Data will be released only in the form of statis- tical summaries. If you have any questions about the survey, please write Ul S = . . . We shall be very grateful for your prompt cooperation. Very truly yours, Ewan Clague Commissioner of Labor Statistics Enclosures 80 U.S. DEPARTMENT OF LABOR BUREAU OF LABOR STATISTICS WASHINGTON 25, D.C. Dear Sir: Several weeks ago we sent you a letter and a set of questionnaires relating to the survey on mathematical employment which is being made by this Bureau. We have checked our responses and find that we have not yet received a report from your company. Your participation in this survey is extremely important to its success, whether or not your organization employs any mathematicians. Like all companies and agencies to which we have mailed questionnaires, your organization is an essential part of a carefully designed sample. We hope, therefore, that you will send us your report as soon as possible. The survey of mathematical employment, which is being made in coopera- tion with the National Science Foundation and the Mathematical Association of America, will obtain information urgently needed in reassessing mathema- tics education. We believe the results of this survey will not only contribute to our national strength and the objectives of the National Science Foundation but will also be of direct benefit to employers of mathe- maticians. ---- As mentioned in our previous correspondence, this survey utilizes two questionnaires, schedule A and schedule B, additional copies of which are enclosed. Please read schedule A first since it contains a statement of the purposes and plan of study and instructions for handling the two questionnaires. All replies will be kept strictly confidential. They will not be published in any way which would permit identification of any company, agency, or individual. Data will be released only in the form of statistical summaries. If you have any questions about the survey or need additional schedules, please write us. We shall be very grateful for your prompt cooperation. Wery truly yours, Ewan Clague Commissioner of Labor Statistics Enclosures 81 U.S. DEPARTMENT OF LABOR BUREAU OF LABOR STATISTICS WASHINGTON 25, D.C. Dear Sir: Thank you for your prompt response to our survey of mathema- tical employment. Your cooperation in distributing questionnaires to your employees engaged in mathematical work is greatly appreciated. In reviewing our returns, we find that we have not heard from a number of the selected employees in your organization. Since the success of the survey depends on a good response from mathema- ticians included in the sample, we should like to solicit your cooperation in urging all your employees who have not yet sent us their questionnaires to do so as soon as possible. We are enclosing additional questionnaires in case they are needed. We shall be very grateful for your continued cooperation. 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