REPORT RESUMES

ED 017 637 VT 000 901 MATHEMATICAL EXPECTATIONS OF TECHNICIANS INMICHIGAN INDUSTRIES, A SPECIAL VOCATiONAL EDUCATIONRESEARCH PROJECT. BY- LAWS, NORMAN G. M1CHIGAN ST. DEPT. OF PUBLIC INSTR., LANSING PUB DATE JUL 66 ED RS PRICE MF60.50 HC -$2.48 60P.

DESCRIPTORS... *SUBPROFESSIONALS, *MATHEMATICS,QUESTIONNAIRES, EMPLOYMENT QUALIFICATIONS, SURVEYS, *EDUCATIONAL NEEDS, INDUSTRY, INTERVIEWS, MICHIGAN,

OBJECTIVES OF-THIS STUDY WERE TO DETERMINE THE FUNDAMENTALS OF MATHEMATICS AND COMMON MATHEMATICSSKILLS NEEDED BY TECHNICIANS AND TO DETERMINE IF THESEFUNDAMENTALS. AND SKILLS SHOULD BE TAUGHT BEFORE EMPLOYMENTOR AS RELATED ACTIVITIES AFTER EMPLOYMENT. A PILOT SURVEY WAS MADEIN DETROIT TO VALIDATE THE RESEARCH INSTRUMENITS.THE POPULATION WAS 71 COMPANIES, SELECTED ON THE BASIS ,..117"BEST PRACTICE" AND "MOST PROMISE" WHICH EMPLOYED 23,0ti2TECHNICIANS. REPRESENTATIVES OF EACH COMPANY WERE IVITZRVIEWED TOASCERTAIN THE TYPES OF TECHNICIANS EMPLOYED AND 'r :ZIRMATHEMATICS NEEDS. THE RESPONDENTS INDICATED ON A CHECKLISTTHE MATHEMATICS FUNDAMENTALS. THEY CONSIDERED ESSENTIAL, DESIRABLE, OR NOT NEEDED. IT WAS ESTABLISHEDTHAT SKILLS-IN ARITHMETIC, ALGEBRA, GEOMETRY, AND TRIGONOMETRY(THROUGH 'RIGHT ANGLES) WERE NEEDED BY ALL CLASSIFICATIONSOF TECHNICIANS. OTHER FINDINGS WERE - -(1) PREPARATIONPRIOR TO EMPLOYMENT WAS DESIRED BY 2T-PERCENT OF THESUPERVISORS, (2) 2.8 PERCENT OF THE COMPANIES OFFERED TRAININGCOURSES, (3) 87 PERCENT OF THE COMPANIES HAD TUITION SUBSIDYPLANS, AND (4) 65 PERCENT PREDICTED THAT, BY 1977, STATISTICS,COMPUTER LANGUAGE, AND CALCULUS WOULD BE REQUIRED OFMANY TECHNICIANS. TABULAR DATA SHOW THE MATHEMATICS FUNDAMENTALSAND SKILLS NEEDED FOR ALL THE TECHNICIAN CLASSIFICATIONSSTUDIED. (EM)

U.S. DEPARTMENT OF HEALTH, EDUCATION & WELFARE OFFICE OF EDUCATION

THIS DOCUMENT HAS BEEN REPRODUCED EXACTLY AS RECEIVEDFROM THE

PERSON OR ORGANIZATION ORIGINAT!N6 IT.POINTS OF VIEW OR OPINIONS

STATED DO NOT NECESSARILY REPRESENT OFFICIAL OFFICEOF EDUCATION

POSITION OR POLICY. Mathematical Expectationsof Technicians in MichiganIndustries

a specialvocational education researchproject

conducted at Wayne State University with assistance from Henry Ford Community College from funds provided for researchand development by Michigan Department of Public Instruction Division of Vocational Education

By Researcher NORMAN G. LAWS, Teacher Henry Ford Community College,Dearborn, Michigan

Under the Direction of G. HAROLD SILVIUS, Professorand Chairman Department of industrial Education,College of Education Wayne State University, Detroit,Michigan

Submitted to ROBERT M. WINGER Assistant Superintendent forVocational Education Department of Public Instruction July, 1966 ACKNOWLEDGEMENTS

The director and the researcher take this opportunity to note the important contribution made by Task Fore One. These business, educational, industrial, governmental, and labor leaders, whose names appear in Appendix C, with their experience and familiarity of Michigan industries, provided recommenda- tions that were most important 'n selecting companies to be interviewed. The high percent of response from industry is evidence of the fine coopera- tion received. The &ector and the researcher were especially impressed by the many executives who so graciously gave of their time and who extended cour- tesies, so that the study could be accurate and complete. Pixies:4°nel services were provided by the faculties of both Wayne State University and Henry Ford Community College. Especially helpful in the areas of their specialties were Dr. Wilhelm Reitz in educational research, 'Dr. Sigurd Rislov in higher education, Professor Gordon E. Rivers in engineering, and Dr. Eugene P. Smith in mathematics education. Individuals from the service areas of both institutions also assisted. The services of Helge E. Hansen (audio-visual director), Carol Waarala (secretary), and Henry Czerwick Jr. (illustrator) are gratefully acknowledged.

NORMAN G. LAWS G. HAROLD SLAWS TABLE OF CONTENTS

Page ACKNOWLEDGMENTS CHAPTER I. A SPECIAL MICHIGAN VOCATIONALEDUCATION RESEARCH PROJECT 1 Introduction 1 Design and Function 1 Methodology 2 Selection of Companies That Were Involved 3

II. PRESENTATION OF FINDINGS 5 Types of Technicians in Michigan Industry 5 Findings From the Interviews 6 The Mathematical Checkoff Lists 11

III. SUMMARY, CONCLUSIONS, OBSERVATIONS AND RECOMMENDATIONS 19 Summary of the Study 19 Conclusions of the Study 20 Researcher's Observations 21 Recommendations for Implementation of the Findings 21

APPENDIXES A. COMPARISON OF MATHEMATICALFUNDAMENTALS __ 22 B. RESEARCH INSTRUMENTS 44 C. PERSONS INVITED TO SERVE ONTASK FORCE ONE 53 D. STANDARD METROPOLITAN STATISTICALAREAS 53 E. MANUFACTURING FIRMSCONSIDERED FOR THE STUDY 54

BIBLIOGRAPHY 56

V CHAPTER I A Special Michigan Vocational Education Research Project

Introduction twin many details, such as a review of the literature, Title of Project not contained in the bulletin (item 4), this study and the doctoral dissertation of Norman G. Laws at Mathematical Expectations of Technicians Wayne State University were planned to complement in Michigan Industries each other. Problem Design and Function A survey conducted by the Presicient's Committee on Scientists' and Engineers indicated that the scope Major Steps in Conducting the Research of mathematics was the most critical determinant for 1. A pilot survey was made of selected and both the level and the quality of technological accessible firms in the Metropolitan Detroit area to curriculums. Probably, the most common criticism validate research instruments. Procedures and instru- among graduates and employers was directed at the ments for data gathering, in the pilot survey, were mathematical content of such curriculums.1 analyzed by the researcher, the director, and the To maintain the ratio of technicians to engineers advisers for needed revisions before conducting the (0:7 to 1), the 775,000 technicians employed in 1960 total project. must increase (521,700 or 67.3 percent) to 1,396,700 2. A task force of selected leaders from govern- by 1970. The increase must be 695,000 yearly by ment, libor, and industry assisted in determining the 1970, if losses due to deaths, retirement, and job firms to be contacted. The members of Task Force changes are considered. Also, if the technician ratio One appear in Appendix C. The firms that were is to be increased (2 to 1), the number of technical surveyed were selected on the basis of their leadership graduates yearly must increase to about 803,300.2 in "best practice" and 'most promise" in respect to The 701 two-year colleges offering technical curricu- the use of technicians, as judged by Task Force One. lums in 1963, represented a 300 percent increase from These companies were chosen from those employing 1950,3 and it seems evident that in the State of Michi- more than five hundred persons and located in the gan the percent of increase will need to be even Standard Metropolitan Statistical Areas of the State greater in the next decade. of Michigan, as defined by the Bureau of the Budget.4 It seemed essential then to determine what funda- 3. Each firm nominated was contacted. A letter mentals of mathematics are necessary for technicians requesting cooperation (as shown in Appendix B) so that existing and future curriculums will better was sent to the president. This letter provided the prepare these students for their place in a techno- opportunity for the president of the firm to designate logical society. the name and title of the person, or persons, with whom the researcher could work. It was anticipated Specific Objectives of the Study that a response at this level would assure cooperation. Another purpose was to provide an opportunity for 1. To determine the fundamentals of mathematics top management of industry to suggest persons, in needed by various kinds of technicians to more effec- their organization, who could help formulate edu- tively perform their jobs in Michigan's industries. cational requirements for technicians in Michigan 2. To determine whether certain common skills industries. in mathematics were needed by all technicians. 4. The researcher then interviewed the designated 3. T3 determine if essential mathematical skills personnel at the firms selected (as shown in Appendix should be taught to the student prior to employment, E) in the metropolitan areas of Ann Arbor, Bay City, or if instruction should be provided as a related Flint, Grand Rapids, Jackson, Kalamazoo, Lansing, activity while the technician is employed. Muskegon, and Saginaw during the summer and fall 4. To prepare manuscript for a bulletin sum- 1The President's Committee on Scientists and Engineers, Re- marizing the findings, conclusions, and recommenda- port on Technical Institute Curricula (Washington: Office of tions of the study. Civil and Defense Mobilization, 1959), pp. 44-46. 5. To use the research information for a project 2Prakken (ed.), Technician Education Yearbook (Ann Arbor, Michigan: Prakken Publications, Inc., 1963), pp. 71-75. of greater depth that will parallel this study. While 3/bid, pp. 9-47. the second study will serve as partial fulfillment of 4Standard Metropolitan Statistical Areas (Washington, D.C.: the requirements for the Ed.D. degree and will con- Office of Statistical Standards, 1964), p. 52.

1 of 1965. Interviews continued with the survey in the set the stage for the second phase of the inquiry Detroit metropolitan area. the mathematical checkoff lists (as shown in Appen- The use of a tape recorder expedited each inter- dix B)0 view and provided the researcher with an accurate Basic to each interview, was the establishment of and complete record so that the information obtained a working definition for a technician (found in Appen- could be analyzed. Data were summarized in the form dix B). While this definition (which is a composite presented as Table 3. from definitions supplied by the Association of 5. Intervieweeswerefurnishedwithenough Manufacturers, the American Society for Engineering mathematical checkoff lists (as shown in Appendix Education, and others) had been sent to the indi- B), and return envelopes, to route a set to each major vidual (or individuals) previous to the interview, the type of technical worker that had been identified. researcher, during the preliminary remarks at each 6. The analysis of data was made with the help interview, explained (for the purpose of this study) of personnel and equipment at the Wayne State Uni- that he was concerned with the high school graduate versity Research Center. Through the use of per- who needed additional training, either formal or infor- centages and ranking categories, conclusions were mal, so that he could assist the engineer, scientist, or drawn, observations made, and recommendations skilled tradesman in doing the more routine engineer- formulated. ing functions in industry and that this definition 7. The manuscript for a bulletin was prepared did not encompass a person who was a certified scien- and released by Wayne State University, in coopera- tist or engineer, or who had obtained a bachelor's tion with the Michigan Department of Public Instruc- degree, as a requircAnent for the particular pay-roll tion, Division of Vocational Education, and sent to assignment. It was further stressed that this study selected persons and institutions interested in the excluded those positions that would be filled by skilled preparation of technicians in the State of Michigan. tradesmen, who had completed a recognized appren- A limited number of copies were sent to additional ticeship program. It was suggested that the technician individuals throughout the United States who have usually performs a broader function than the skilled been identified as currently interested in educational tradesman. Interviewees were appraised, also, of programs to prepare technicians for industry. twenty-two sample technician titles (as noted in Appendix B) that seem to be popular. Personnel and Facilities for the Research With this background, the question was asked, Dr. G. Harold Silvius, Professor and Chairman, "What types of technicians do you employ in your Department: of Industrial Education, College of Edu- company?". The basic purpose of this question was cation, Wayne State University, directed this research. to ascertain the number of mathematical checkoff lists The researcher was Norman G. Laws, instructor at that should be left with the company. It was then Henry Ford Community College in Dearborn. As the requested that the immediate supervisor or senior need arose, other members of the Wayne State Uni- technician, for each type of technician identified, fill versity faculty were consulted in areas of their respec- out the mathematical checkoff list (shown in Appen- tive specialties. dix B) to indicate the elements of mathematics that The Henry Ford Community College provided the workers under his supervision and direction, use the necessary office facilities for the researcher, and in their day-to-day operation. Wayne State University provided office and consulta- As an additional clue to the sophistication of the tion space for the director. Both institutions made technician being employed in a particular company, available the needed office equipment, furniture, and the interviewee was asked about the work experience machines. and education of technical personnel. This was to Personnel 'at the service facilities of the Audio- determine if personnel doing such work were usually Visual. Departments, Computer Center, and Printing reassigned within the company, and, if so if it was Department at both Wayne State University and thought that this was an indication that the needed Henry Ford Community College, were called upon post-high-school preparation was largely a matter of for assistance. on-the-job experience. When this was not the case, it was then assumed that the technician on the job could have been the product of a post-high-school Methodology institution. The question also provided a clue as to what type of institutions were supplying needed tech- Interview Procedures nical personnel. Each interviewee was asked to further The basic purpose of the interview was to ascer- indicate the basic amount of mathematics that he tain company policy in the use of technicians. A copy thought all technicians in his company needed. It of this interview instrument appears in Appendix B. was recognized that the response would be much more Persons in a position to actually hire personnel, or general than that which would later be received from to set up guide lines for such hiring, were interviewed the mathematical checkoff list, but these data were as to their mathematical expectations for technicians. considered significant in determining requirements of Questions were devised to discover what management employers. Thus, the purpose of the interviews was thought were the mathematical requirements of tech- to determine, from the viewpoint of management, the nicians, whom they had employed, or would employ amount and types of mathematics that the technician in the future. This initial interview was planned to needed, to perform the job.

2 The inquiry further pursued the question of In 1949, the Bureau of the Budget of the Executive whether or not the technician should be taught all Office of the President of the United States defined basic mathematical skills while in school, or if he the Standard Metropolitan Areas as the Standard should continue to learn mathematics while on the Metropolitan Statistical Areas of the United States, job. This was done by inquiring about the mathe- to replace four different sets of definitions that had matical deficiencies that had been noted among tech- been in use for various governmental statistical nical workers. Then there was an inquiry into company studies. Thus, this new designation replaces the pre- encouragement for additional schooling through incen- vious designations of Metropolitan Districts, Metro- tive programs such as tuition refund, in-service train- politan Counties, Industrial Areas, and Labor Market ing programs, and motivation efforts to encourage Areas. The primary objective was to standardize the the worker to extend mathematical competency. In- definitions of metropolitan areas, and to make it pos- terviewees were asked, also, about the mathematical sible for all federal statistical agencies to use the same needs of technicians in the decade aheae. boundaries in publishing statistical data.5 The gen- eral concept of a metropolitan area is one that is 'The researcher then provided an opportunity for integrated economically and socially with a recognized the interviewees to make specific recommendations large population nucleus. Thus each Standard Metro- relative to curriculums in technical mathematics. politan Statistical Area must contain at least one city Persons were asked if there were basic topics in mathe- of 50,000 inhabitants. A Standard Metropolitan Sta- snatics that (1) should 'be given more emphasis, (2) tistical Area includes the county of such a central city should be included or excluded, and (3) if there were and adjacent counties as are found to be metropolitan new types of mathematics, especiallythose involving in character and economically and socially integrated computers and new industrial approaches, that should with the county and central city. Actually such an be added. 'Finally, they were 'asked if they had addi- area might contain more than one city of50,000 popu- tional recommendations relative to the way mathe- lation, with the largest city considered the nucleus matics for technical students should be taught. and is often the name given to the area. Tri have met- ropolitan character, at least seventy-five percent of Mathematical Checkoff List Procedures the labor force of the county must in nonagricul- The mathematical checkoff lists provided an op- tural labor.° portunity for technical workers (or their immediate 'In the State of Michigan (as indicated in Appendix supervisors) to describe in greater detail the mathe- D), there were ten such designations. These included matical needs of the technician. It was felt that the Ann Arbor, Bay City, Detroit, Flint, Grand Rapids, worker could best judge the mathematics involved in Jackson, Kalamazoo, Lansing, Muskegon Muske- the daily operation of his job. . gon Heights, and Saginaw. Ann Arbor is adesignate 'There were six subject sections on each form: for the City of Ann Arbor As well as Washtenaw Arithmetic, Algebra, Geometry, Trigonometry, Cal- County, with a population of 172,440, The BayCity culus, and Additional Mathematics. Within each cate- area included Bay City, as well asBay County, with gory, the various fundamental skills werelisted. a population of 107,042. TheDetroit district included These had 'been worked out carefully with members Macomb County, Oakland County, and Wayne of the Henry Ford Community 'College Mathematics County, as well as twenty-three major cities, with a Department and reviewed by Dr. Eugene P. Smith of total population of 3,762,360. The Flint a-eaincluded the Department of Mathematics Education at Wayne the City of Flint, as well as Geneseeid LaPeer State University. Within each category, space was Counties, with a total population of 416,239. Grand provided for additional comments, but few were made. Rapids included the City of Grand Rapids, the City of Wyoming and Kent County, and Ottawa County Most of the forms were returned promptly but a for a total population of 461,906. Jackson included few required follow-up. After. asecond reminder, a the City of Jackson, and Jackson County, with a telephone call was made and finally 91.7 percent of population of 131,994. Kalamazoo included Kalama- the forms were returned (11 ,few more were received zoo County, and the City of Kalamazoowith a popu- too late for classification). lation of 169,712. The Lansing area encompassing Lansing and East Lansing, as well as Clinton County, Selection of Companies That Were Involved Eaton County, and Ingham County had 298,949 per- sons. The Muskegon areaincluded Muskegon end Selection of Areas Muskegon Heights, and Muskegon County, for a total population of 149,943. And Saginaw, included Early in the planning for this study, it was deter- for mined that a random sample approach, in surveying the Saginaw City and the Saginaw County areas industry to determine the mathematical needs of a population of190,752.7 technicians, might not be enlightening. What the investigators were concerned about were those indus- tries, and those companies, which were showing lead- 5Standard Metropolitan Statistical Areas, (Washington, D.C.: ership in !`best practice" and "most promise" as these Office of Statistical Standards, 1964), pp. vii-viii. factors relate to the use of technicians. It was thought elbid. that this could best be accomplished by surveying 7Standard Metropolitan Statistical Areas, Executive Office of selected larger companies in the more highly indus- the President, Bureau of the Budget, Kermit Gordon, Director, trialized areas of the State. Office of Statistical Standards, 1964, pp. 4-42. 3 It was determined als ) by Task Force One that Identification of Companies thirty-six of the listings were actually divisions of Actual companies located in each statistical area larger companies so they recommended that these had to be identified "y size and product. Boards or smaller units be included in the interviews conducted Chambers of Commerce in each of the major cities at the corporation level. These companies were Amer- involved were contacted and they furnished lists of ican Motors, Brunswick Corporation, Consumers companies, within their own areas, that employed 500 Power Company, Chrysler Corporation, Ford Motor er more persons. From these reports, amaster list Company, General Motors Corporation, R. C. Mahon of 205 companies was designated. These were then, Company, and Michigan Consolidated Gas Company. the Michigan Manufacturing Companies located in The interviews in these companies were usually mul- the Standard Metropolitan Areas of Michigan and tiple, were at the corporation level, and reflected the employing more than 500 persons that Task Force results of composite consideration within the com- One considered, for the purpose of nominating com- pany. panies to be invited to participate in the study. This list of Michigan companies appears in Appendix E. Stratified Population Thus 176 firms constituted the stratified popula- Type of Industry tion that was considered by Task Force One. Seventy- To increase the validity of the selection, Task nine companies were actually selected for the study Force One checked the type of industry, in the cate- on the basis of "best practice" and"most promise" in gories of construction, ferrous metals, nonferrous their use of technicians. This constituted 44.8 percent metals, fabricated metals, non-electrical machinery, of those which appeared in the listing in Appendix electrical machinery, motor vehicles, chemicals, com- E. Fifty-two of the companies, or 25 percent were munications, utilities, engineering and architectural rejected by Task Force One because the type of prod- services, public administration, and miscellaneous. uct manufactured was not conducive to hightechni- When it was noted by Task Force One that there cian involvement. was a lack of representation in some ofthese areas, Once 'the companies were selected, letters, like the it Was suggested by the members of this group, that samples in Appendix B, were sent to top administra- certain technical societies be approached for their tors seeking participation. The response (98.8 per - recommendations for participation. Thus, The Engi- cert) was very favorable. Persons contacted were most neering Society of Detroit, the National Association cooperative in both the oral and written phases of the of Engineering Companies, the Detroit Tooling Asso- inquiry. It would be unfair to say that the others ciation, The Association of General Contractors of were uncooperative. Such reasons 'as"the company is America, the Builders Association of Metropolitan moving," "a recent turnover of top personnel," and Detroit, and The American Institute of Architects "company on strike" were given for nonparticipation. were contacted relative to their useof technicians. Due to decentralization, some of the larger companies Subsequently, it was determined that the builders required as many as ten separate interviews, while associations and the general construction companies other companies were in a posiVon to set up area con- utilized apprentices, but not technicians as defined ferences in which persoLanel from various parts of the for this study; that architects associated with the company took part. In all ofthe larger companies, American Institute of Architects did not involve tech- multiple personnel were involved in the interviews. nicians in their work. The tooling associations and Usually the president or personnel director inthe engineering societies also indicated that they did not smaller companies was in a position, by himself, or have a high usage of technicians. So the remaining with the aid of one or two men, to provide the needed technical societies accounted for the additionsto the information. In all cases the checkoff lists were routed basic list. This had the added advantage of adding beyond the interviewees to persons closer to the actual seven smaller firms to the study. technical occupation.

4 CHAPTER II

Presentation of Findings

appliances, end Michigan Industry include business machines, 'household Types of Technicians in even ships. In 1960there were 2,030,282 industrial The colorful era of 'the lumberjack and timber workers in the ten standardmetropolitan areas with baron, in Michigan, elidein the 1801's due to the 40 percent of them involvedin manufacturing.3 depletion of Michigan fore:.cs, 'butwood products are The number of technicians used inMichigan in- still manufactured in Bay City,Escanaba, Ludington, dustry in 1960, in rank order, isreported in Table 1. and Traverse City. Kalamazoois known for its pro- This has been compared to thenumber of companies, duction of paper, but many of thewood raw materials in each industry, that weresvxveyed. The plan wa. are now imported.Thus, Michigan no longerdepends to make more frequent contactsin those industries upon timber as itsprime natural resource,and com- that had the 'highest use oftechnicians. Although it panies, like Lufkin Rule inSaginaw, have survived by might seem that fewer technicians wererepresented finding new products and markets.It was the timber from the motor vehicles industrythan the machinery resources, and transportationfacilities, that origins. .4 industry, it must be remembered thatcompanies rep- drew the carriage works to theState. These were later resenting the former were quite large. converted to automobile plants.Now, over half of the employed in the In the communications andutilities industries there industrial workers in Michigan are was indication of anemerging need for technicians, transportation industry.1 but in the printing industryit was found that the Workers in MichiganIndustries work was handled by thetradesman. During the two world wars, the greatautomotive Types of Technicians in Study plants in Dearborn, Detroit, Flint,Kalamazoo, Lan- tanks, and fatly technicians were found inMichigan indus- sing, and Pontiac manufactured guns, try but few companiesactually use the word :echni- other products for war. In1966, the automotive in- clan in the job title. Thus,these positions were iden- dustry is of major importance,but Michigan is a tified primarily by the job description educational highly diversified industrial state.Just as Detroit is prerequisites. Most employers wereable to classify known as the automotivecapital, Grand Rapids is Battle Creek is their workers after looking ntthe list of descriptive famous for furniture production, titles (reproduced in AppendixB). known for cereal, Midland andLudington are great chemical producers, while AnnArber, Detroit, and TABLE 2 the drugs they manu- Kalamazoo are world known for NUMBER OF TECHRCRNS facture. Cities like Detroit,Dowagiac, Hastings, and Saginaw rival other cities inthe nation in the produc- SURVEYED BY TECHNOLOGIES tion of iron and steel. Michigan'sindustrial products 59 drafting 20 chemical 8 civil 46 mechanical18 instrumenta- 5 fluid power TABLE 1 38 laboratory tion 4 packaging IDENTIFICATION OF MICHIGAN 31 electrical & 14metallurgical 2 aeronautical 1 communica- TECHNICIANS BY TYPE OFINDUSTRY' electronic 9 data process- 26 industrial ing tions Tedmicial4 Employed Number of Different When the mathematical checkofflists were re- By Industries Industries Surveyed ceived, those job titles that had no',been labeled by 9961 motor vehicles 20 machinery a technology wereclassified by reviewing remarks 4035 machinery 15 motor vehicles made in the recorded interviews.Table 2 is a listing 2715 ferrous & fab. 8 chemicals of the technologies with anindication of the number metals 5 electrical machinery of returns in each category.One of the telephone 1910 chemicals 5 ferrous & fabricated companies insisted that theirtechnicians must prop- 1328 construction metals erly be called communicationtechnicians although 1070 electrical machin- 2 construction another telephone company usedthe electronic tech- ery nician classification. Moreoften the broader classifi- 2 furniture instead of fluid power, 462 non-ferrous metals 2 communication cations, such as mechanical 317 furniture 2 utilities 1U.S. Bureau of the Census, U.S. Censusof Population: 1960, 1 non-ferrous metals Vol. 1, Characteristics of thePopulation, Part 24, Michigan, 1264 utilities (U.S. Government Printing Office,Washington, D.C., 1963), miscellaneous 9 miscellaneous pp. 582-592. 2/bid. 23062 Total 71 Total 3/bid. 5 *Data Processing 4.2%Writer 1.3% Fluid Power 1.6%Aeronautical 0.6% Civil 1.9%Communication 0.6% Packaging 1.6%Biological 0.3% FIG. 1 TECHNICIANS SURVEYED were used. The technology classificationseemed to technicians. Of the supervisors interviewed, 50.7 per- reflect the 'area of preparation more than the specific cent had been employed with the same company for job assignment. more than fifteen years, while 42.2 percent hadbeen Figure 1 shows the relationship of the numbers of in their present position for more than ten years technicians surveyed in each technology by percent- (Table 4). Thirty-one percent of the supervisors were age. Although the investigation 'attempted toreflect 46 to 50 years of age; only 11.2 percent were younger. the actual technologies in proportion to their usage, While 81.7 percent have college degrees (Table 5), it may be noted that the seven major classifications 52.1 percent have only a bachelor's degree. Further, represent a fairly even distribution. although only 59.2 percent graduated from colleges of engineering, 67.6 percent report (Table 6) having Findings from the Interviews had calculus in their mathematical background. Types of Personnel Interviewed Technician Evaluation by Supervisors 'The chief administrative officer of each firm, as was mentioned, appointed the person or personsto While almost every firm indicated that they would be contacted. Tables 3 through 10 provide a summary like to employ technicians who had completed two of the interview responses made at seventy-one com- years of college work, 46 percent indicated that they panies. This summary also represents many confer- had and were willing to accept high school graduation ence sessions and multiple interviews atindividual as a minimum educational requirement(Table 7). companies. No weight factors were given (by size of Thirty-five percent now require an associate degree, company) since it did not appear that patterns devel- or equivalent, as a minimum educationalrequirement oped according to size of company. for their technicians. Private trade schools, service Of those interviewed (Table 3), 35 percent were schools, and technical institutes provide the source of managers, 18 percent presidents orvice-presidents, many of the present technicians. Some employers seek and 13 percent directors. Thus, 66 percent of those the two-year college dropouts, but often there appears interviewed were at a high level of management. Per- to be a feeling that this type of person represents an sonnel directors accounted for 15 percent, and re- engineer who is not fully qualified, as a technician. searchers for another 6 percent of the contacts. Eleven Many employers admit that they have used engineers percent were chief mechanics or engineers. All of these to do work at the technician level. persons provided much essential data sincethey had Fifty-nine percent of those contacted were sads- a great deal to do with company policyin the use of fied with their present technicians (Table 7) while

6 TAD= 3 POSITION AND MPS. KU COMPANY

Position With Company . Years With Company

k 42 14 0 Industries Surveyed a k k UN 0 IM 0 k 0 0 S i b 49 6-1 N N 10 4-1 01 ni 6 4) 11 0 0 0 .4.1 00 .1 toot; I. 0 0 0 40 .0 0 0 0 0 0 El a 0 0k k 0 .C1 0 0 ; ft 6 vl 0 4) Xvi It A A 41 AX° 1:14 A 10 0 4-1 1 3 4 5. 4 2 1, Machinery 3 2 7 2 2 1 2 1 2 2 4 3 3 1 Motor Vehicles 2 4 2 2 1 1 2 1 3 2 1 5 1 2 2 2 1 1 Electrical Machinery 2 1 2 1 1 2 2 Ferrous & Fah. Metals 1 1 2 1 1 1 Construction 1 1 Furniture 1 1 1 1 Communications 2 1 1 Utilities 1 1 1 1 Non-ferrous Metals 1 1 2 2 1 1 1 1 Miscellaneous 1 2 1 1 2

4 1 8 11 1015 13 8 6--- Total 6 7 25 9 6 2 11 5.6 1.4 32.315.514.721.118.311.38.5 Percent 8.5 9.9 35.2 12.7 8.5 2.8 15.5

TABLE 4 TSARS IN POSITION AND AGEOF SUPERVISORS INTERVIEWED

Years In Position Age

g Industries Surveyed k IA UN I 1/40 UN 0 X 1) R a 0 4-1 N N 3 5 UN 4-1 o o 0 o fro ig El El 4° co HE4 E E4 li 0 0 El to MD 4-1 .0 E4 El 1A 0 co 4-1 4-1 42 ri 14 146 4' -I* IA '0 a 0 q) 1 4-1 N N Z 20 2 2 1 1 4 7 5 3 Machinery 5 5 5 4 15 1 1 1 1 2 5 3 Motor Vehicles 3 2 4 3 3 8 1 1 2 2 1 Chemicals 6 1 1 3 1 5 Electrical Machinery 1 1 2 2 1 1 2 2 5 Ferrous & Feb. Metals 1 1 1. 1 1 2 Construction 1 1 1 2 1 1 Furniture 1 2 1 1 Communications 1 1 1 1 1 1 3 Utilities 1 1 1 .1 Non-ferrous Metals 1 4 1 8 1 1' 1 2 Miscellaneous 4 2

6 2 .12 22 16 13 71 21 14 13 8 5 4 6 Total 18.3 19.8 18.3 11.3 7.05.68.5 8.5 2.8 16.9 31.0 22.5 Percent 29.6

7 TABLE $ EDUCATIONAL BACKGROUND 07 SUPERVISORSINTERVIEW=

Type of College Attended Major Field of Study

to to Industries Surveyed k I; 0 k 0 M k 0 0 m a .r, % Ig e e 0 m A 01 a a k g O O w 41 44 a 4 r1 k 4 4 A O U A A .+P o O A 0 0 .14 z o z t) A) El Z 0 01 ir 0 2 1 14 2 1 Machinery 12 1 2 1 3 1 1 11 1 1 Motor Vehicles 6 3 2 3 1 1 2 2 4 Chemicals 4 1 2 1 1 4 Electrical Machinery 3 1 1 1 2 Ferrous & Fab. Metals 2 1 2 2 2 Construction 1 1 1 1 Furniture 2 2 Communications 1 1 1 Utilities 1 1 1 1 Non-ferrous Metals 2 3 3 Miscellaneous 3 3 2

8 4 6 42 3 1 7 Total 32 10 3 5 14 7 8.5 59.2 4.21.4 9.9 Percent 45.114.14.2 7.019.79.911.3 5.6

?AILS 6 'INTERVIEWED ACADEMIC STATUS AND MATRINSTICALBACKGROUND OP SUPERVISORS

Academic Stgtua Mathematics Background o /-1 to to .0 k 10 4.) 4 M N 0 Industries Surveyed 10 4 X 0 0 4 k X 14 A 0 44 0 14000 000 0 0 m0 .0 0 AN ooa k g) 0El -Pk +Ik tO 00 4 to0 0 i 0 e:1 4 gt A 0 43 il °63 AA: 10 2 1 Machinery 5 12 2 1 7 2 8 4 1 Motor Vehicles 1 3 8 2 1 4 2 1 1 Chemicals 4 2 2 1 1 Electrical Machinery 1 3 1 1 1 1 2 Ferrous & Fab. Metals 3 2 1 1 1 Construction 1 1 1 1 Furniture 1 1 1 1 Communications 2 2 1 Utilities 1 1 1 1 Non-ferrous Metals 1 1 3 2 1 1 Miscellaneous 2 4 1 1

27 13 6 4 Total 3 18 37 2 5 6 21 29.638.0 18.3 8.5 5.6 Percent 4.2 25.4 52.1 2.8 7.0 8.5

8 MU 7 CAW SlIPZINISOIE IIIMITZW121 EDUCATIONAL DACEODUED0! Ma MOWS AD

Educational Background Mathematical Background

Industries Surveyed I 0

Machinery 9 5 6 Motor Vehicles 6 7 2 Chemicals 4 4 Electrical Machinery 2 2 1 Ferrous & Fab. Metals 1 1 3 Construction 1 1 Furniture 1 1 Communications 2 Utilities 1 2 Non-ferrous Metals 1 Miscellaneous 6 1 1

42 22 1 1 4 1 Total 33 9 25 1 3 59.2 31.0 1.4 1.4 5.6 1.4 Percent 46.5 12.7 4.2 35.2 1.4

MU 8 ADVOCATED EY SUPERVISORSINTERVIEWED MATHEMATICALMILLS OT TECHNICIANS Rule Computer Requirements Mathematical Emphasis Skill Skill Skill In Ten Years

,..) 0 4., x Industries Surveyed ?, 0 A 0 ik * 41 M o 10 . 0 o o 1 d 0 o z VI H it A4 H al 7 13 2 13 19 1 9 7 4 Machinery 13 3 7 8 15 7 4 4 5 10 Motor Vehicles 14 1 7 1 7 1 3 5 3 5 Chemicals 8 7 4 2 5 3 1 1 1 Electrical Machinery 4 5 3 2 5 3 1 1 2 3 Ferrous & Fab. Metals 4 1 3 2 2 1 1 2 Construction 2 2 2 1 1 1 1 Furniture 1 1 1 2 2 2 1 Communications 2 2 1 3 1 2 3 Utilities 2 1 1 1 1 Non-ferrous Metals 1 1 2 6 1 2 6 8 4 4 Miscellaneous 6 1

25 47 33 38 69 2 34 21 16 Total 57 5 10 b5.2 46.553.597.22.848.029.522.534.8 Percent 79.2 6.9 13.9

9 TALI f BASIC JUITEINATICS 10 TiCEINVAIISADVOCATED DT SE1111118011 IIITLRYIZWID Basic Mathematical Ability Math. Skill qualifi- Mathematics Needed Compared to Engineerscations For Employment_ rft 03 4) s,b '0 rl k H m 43 0 .,4 qg :I it Z 1 0 0k Industries Surveyed 40) tills 1 X .4 ril 42 rlI 43 ej .1 it t 0 44 V 0 te 0 M 0 0 3 i g .s4 0 43 ri 00 0 ri ri M k 0 0 0 8 H 8 0 RN H 0 E4 13 1 Machinery 1 2 17 18 2 6 11 1 Motor Vehicles 2 1 12 15 3 7 Chemicals 1 6 1 8 1 Electrical Machinery 1 4 5 5 Ferrous & Fab. Metals 1 4 5 2 3 2 Construction 2 2 1 1 Furniture 2 2 1 Communications 1 1 1 1 1 Utilities 3 3 3 Non-ferrous Metals 1 1 1 Miscellaneous 3 5 8 3 5

24 44 2 1 Total 4 6 3 57 1 68 3 4.2 33.8 62.02.8 1.4 Percent 5.68.5 4.280.3 1.4 95.8

TAILS 10 ADVOCATED DV V180113 INTERVISEED CONTINUED MATICOOLTICALDEVELOPNENT

Continued Formal Schooling Additional Mathematical SkillsNeeded 4 e o111 CO 00 V H P4 Industries Surveyed , 11 14 0 3 0 14o OS 11 1 3 .403 er Pi 0 40 o i o 0 0 0 013 4 et 00 0PI E4 M Z 19 1 Machinery 2 1 13 2 Motor Vehicles 1 3 8 Chemicals 4 1 Electrical Machinery 1 4 1 Ferrous & Fab. Metals 2 Construction 1 2 Furniture 1 1 1 Communications 1 3 Utilities 1 1 Non-ferrous Metals 7 1 Miscellaneous 1

62 2 6 1 Total 3 1 5 3 87.3 2.8 8.5 1.4 Percent

10 31.0 percent felt that their background in mathe- received an essential rating. Other geometric concepts matics, should have been more practical. This idea was were listed as desirable 'but interestingly,the men- followed by a strong recommendation by 79.2 percent suration formulas were rated as not needed by many. (Table 8) that practical mathematics receive more Analytical coordinate geometry was also judged as emphasis intechnicalmathematicscurriculums. not needed. Forty-six percent listed calculator skills, 97.2 percent There was a strong indicated need for right tri- slide rule skills, and 48 percent computer skills as angle trigonometry. Topics dealing With angles, func- useful for technicians. Sixty-five percent predicted tions, and tables were considered essential, but the an increased mathematical need for techniciansin the solution of oblique triangles and analytical 'trigo- next ten years. nometry was considered only desirable. Mathematical ability through trigonometry for No calculus skills were considered important to technicians was suggested by 80.3 percent of the the work of the technician. Few topics in the area even supervisors (Table 9). Almost all(95.8 percent) received a desirable classification and most were agreed that the technician needs less mathematics marked as not needed. than the engineer, but 4.2 percent emphasized that Unexpectedly, the ability to use a calculating the technidian needs a more practical mathematics. machine was considered a desirable skill for a tech- While 33.8 percent of the supervisors expected the nician. technician to know his mathematics before being employed, 62 percent recognized that the 'application Mathematical Fundamentals by Technologies Phase of the technician work would provide for growth In addition to a comparison of the mathematical in mathematical ability. It was encouraging to note fundamentals needed by all technicians, as shown in that 87.3 percent of companies involved in the study Figure 3, it seemed desirable to compare these funda- had plans that encourage employees to continue their mentals by technologies. The essential, desirable, and education (Table 10). not needed responses by technicians have been sum- Figure 2 indicates in percentages the amount of marized in Appendix A. In many responses there was mathematics that supervisors in various industries almost total agreement 'but some interesting excep- feel that their teekrIciails should have. The center tions were noted. pie of Fizure 2 is a summary of the individual cate- While 'logarithms (at least common logarithms) gories and clearly shows that trigonometry is the were rated as desirable for most technicians indus- most advanced course advocated. trial, laboratory, mechanical, and packaging techni- cians indicated that logarithms (of any kind) were The Mathematical Checkoff Lists not needed. Only the aeronautical, drafting, mechani- cal, and fluid power technicians felt that mathematics All Technicians dealing with threads and gears was essential. The items 'on the mathematical checkoff lists re- Some types of technicians deal basically with turned were compared (by technician type, and by arithmetic calculations and simple algebra including industry type) to determine areas of agreCilient and linear equations. Thus, chemical, industrial, instru- disagreement. These comparisons are reported in mentation, laboratory, mechanical, and metallurgi- Figure 3, in terms of percent of response in cate- cal technicians rated few algebraic topics as essential. gories of essential, desirable, and not needed. The drafting, electrical, packaging, fluid power, and Although some of the mathematical fundamental communication technicians decided that systems of skills are more important than others, Table 11 shows equations and quadratic equations were essential that the fundamental operations of arithmetic, com- fundamentals, Only the aeronautical and data pro- mon and decimal fractions, significantfigures, stand- cessing technicians rated the most advanced algebraic ard notation, ratio and proportion, and use of tables topics as essential. were the most basic skills needed. Weight and meas- Civil, drafting, electrical, fluid power, and con- urement conversions, multiplication and division on struction technicians were the only ones to give the slide rule, measurement, and micrometer reading emphasis to more than the very basic elements of were considered important, but fewtechnicians need geometry. The metallurgical technicians, especially, the machinist mathematical skills of indexing, gear had no need for the mensuration formulas. ratio, thread and screw formulas. Also, few felt that Beyond right triangle trigonometry, civil, drafting, the ability to use the slide rule to solve problems of and mechanical technicians ranked the solution of powers, roots, and percentage was essential. oblique triangles as essential but only aeronautical, Basic algebra from the concept of real numbers, drafting, mechanical, laboratory, metallurgical, and the four fundamental operations, signs of aggrega- civil technicians, rated analytical trigonometry as tion, absolute values, ratio, proportion, and fractions, even desirable. factoring through linear equations was considered Aeronautical, data processing, electrical, and pack- essential for technicians. But inequalities, functions, aging technicians indicated that some practical cal- systems of equations and graphic representation culus would be desirable. As would be expected, data seemed only desirable while determinants, high order processing technicians indicated that knowledge of equations and other advanced topics were rated as analog and digital computers was essential for them, not needed. but only aeronautical, chemical, civil, electrical, pack- Basic figures and geometric facts about lines, aging, fluid power and communication technicians angles, and planes were the only fundamentals that even classified them as desirableskills.

11 NON-FERROUS FURNITURE UTILITIES CONSTRUCTION FERROUS METALS METALS

MACHINERY FASIMATED METALS ALL TECHNICIANS

ELECTRICAL MOTOR VEHICLES COMMUNICATIONS MISCELLANEOUS CHEMICALS MACHINERY

FIG. 2 - NEEDED MATHEMATICAL SUBJECTS AS REPORTED BY SUPERVISORS iN MAJOR MICHIGAN INDUSTRIES

Mathematical Fundamentals munications, listed algebra fundamentals as essential at least through quadratic equations. by Type of Industry In geometry the pattern changed some. Techni- A comparison of the mathematical needs of tech- cians in non-ferrous metals and electrical machinery nicians can be made according to the type of industry industries needed less skill in this subject, while tech- involved. This comparison provided a greater variety nicians in construction, motor vehicles, utilities, and in the responses. Even in the section dealing with communications industrieslisted more geometric arithmetic skills there was found a difference in im- fundamentals, as essential. portance when this type of comparison was made. Trigonometry was judged as not needed by tech- Appendix A shows that technicians in construction, nicians in the non-ferrous metals, electrical machinery, electrical machinery, furniture, and fabricated metals and furniture industries, but 'at least desirable by industries require less arithmetic skills than those in most, and essential for construction, ferrous metals, ferrous metals, non-ferrous metals and communica- and utilities. An understanding of right triangle trigo- tions. Except for a lack of demand for plain and dif- nometry was felt as essential, or desirable, by all ferential indexing, complex gear ratio, lead screws, but the demand for oblique triangle trigonometry and natural logarithms, the arithmetic fundamental and analytical trigonometry was limited to construc- skills were deemed ,desirable for all. The desirable, tion, ferrous metals, and utilities industries. rather than esuential, classification was most fre- quently indicated for percent, interest, and common No calculus was suggested as essential, bit tech- logarithms, especially as calculated on the slide rule. nicians in fabricated metals and communications Although the more elementary topics of algebra industries indicated that basic calculus skills were were listed as desirable, technicians in construction desirable. and furniture industries indicated no need for this 'Among technicians in construction, motor vehicles, subject. Even technicians in fabricated metals, and communications, and utilities industries, the analog electrical machinery inde Aries showed a limited need and digital computer language skills were judged for algebra by rating content through simple equa- desirable. All but electrical machinery rated calcu- tions and formulas as important, and the more ad- lating machine skills as desirable and fabricated vanced skills as only desirable. But the other industries metals, chemical, and the furniture industry rated including ferrous metals, motor vehicles, and coin- these skills as essential.

12 TABLE 11 MATHEMATICAL FUNDAMENTALS BY RANK ORDER Arithmetic Skills Algebraic Skills Geometric Skills Essential: Essential: Essential: Fundamental Operations Fundamental Operations with Lit- eral Numbers Basic Geometric Figure Concepts (addition, subtraction, multiplica- Lines, Angles, Planes tion) (addition, subtraction, multiplica- tion) Fractions, Common and Decimal Desirable: Ratio, Proportion, Variation Standard Notation and Significant Congruence Figure Fractions, Fundamental Operations Measuring (rule, micrometer, ver- Formulas and Linear Equations Polar Coordinate Functions nier) Signs of Aggregation (parenthesesSimilarity Tables and Interpolation etc.) Constructions Ratio and Proportion Explicit and Literal Numbers Parallelism in Lines and Planes Converdions (weights and meas- Absolute Values ure) (English and Metric) Symbolic Expressions Not Needed: Approximation and Estimation Desirable: Locus Slide Rule, Multiplication and Div. Complex Fractions Inequalities Slide Rule, Ratio and Proportion Factoring and Special Products Mensuration, Plane Percentage and Interest Use of Functions and Relations Mensuration, Solid Desirable: Inequalities Coordinate Geometry (analytic) Graphical Representation of Equa- Methods of Proof and Reasoning Common Logarithms tions Slide Rule, Powers and Roots Quadratic Equations Slide Rule, Percent and Interest Systems of Equations Trigonometric Skills Slide Rule, Common Logarithms Essential: Not Needed: Not Needed: Natural Logarithms Progressions, Arithmetic Angles and Coordinates Slide Rule, Natural Logarithms Exponential Equations Trigonometric Tables Gears and Gear Ratio Progressions, Geometric Trigonometric Functions Screw Threads and Spirals Vector Algebra Right Triangles, Trigonometric Solution of Plain and Differential Indexing Determinants Complex Gear Ratio and Lead Synthetic Division Screw Equations of Any Power Desirable: Complex Numbers Slide Rule, Trigonometric Func- Calculus Skills Higher Order Equations tions Not Needed: Binominal Theorem Variables, Functions Number Theory Not Needed: Limits Additional Oblique Triangles, Solution of Basic Theorems Mathematical Skills Trigonometric Functions by Arc Integration Length Method Differentiation Essential: Trigonometric Functions by Unity Operation of Calculating Machines Integration Formulas Method Methods of Integration Not Needed: Plane Vectors Polar Coordinates Probability Periodic Functions Infinite Series Applied Statistics and Dynamics Graphs of Trigonometric Functions Vector Analysis Analog and Digital Computers Analytical Trigonometry, Formu- Parametric Equations Curve Fitting las Trancendental Functions Vector Analysis Analytical Trigonometry, Equa- DiffereTial Equations Logic Systems tions Partial Differentiation Root Approximation and Identifi-Analytical Trigonometry, Identi- Hyperbolic Functions cation ties Slide Rule, Hyperbolic Functions Boolean Algebra Complex Numbers Multiple Integrals Set Theory DeMoivre's Theorem

13

.-- Fig.3:-- MathematicalFundamentals For AllTechnicians %Not Nieded %Essential %Desirable ARITHMETIC SKILLS

Fundamental Operations Fractions, Common And Decimal Standard Notationand Significant Figures Slide Rule, Multipli- cation and Division Slide Rule, Ratio And Proportion Slide Rule,Powers And Roots Slide Rule,Common Logarithms Slide Rule,Natural' 46.6 Logarithms Slide Rule, Percent 28.8 And Interest Approximation and .9 Estimation Conversions

Measuring Tables and Interpolation Percentage and 25.1 Interest Ratio and Proportion 21.1 Common Logarithms 41.9 Natural Logarithms 1 Plain and Differen- 65.3 tial Indexing Gears and Gear.Ratio

Screw Threads and 47.2 Spirals 'Complex Gear Ratio 65.4 And Lead Screws

14 Fig. 3. - -Continued EBRAICSKILLS

Explicit and Literal 17.5 NUmbeis 2.5 Symbolic Expressions Fundamental Operation 0.0 With Literal Symbols 13.1 Signs of Aggregation 18.4 Absolute Values Ratio, Proportion, 8.9 Variation Fractions, Fundamen- 9.6 tal OperatiOns 26.4 Complex Fractions Factoring and Special 26.5 Products Formulas and Linear 0.3 Equations 32.7 Inequalities Use of Functionsand 29.4 Relations 4o.4 Systems of Equations 59.8 Determinants 47.2 Quadratic Equations Higher Order 28.1 65.1 Equations Exponential 56.4 Equations Graphical Representa- 35.6 tion of Equations 62.3 Synthetic Division

Equations of Any 62.6 1 Power Complex NUmbers 63.3 1

Binominal Theorem 64.7 Progressions, 52.7 Arithmetic Progressions, 58.2 Geometric Number Theory 24.4 68.8

Vector Algebra 59.8

15 Fig. 3.--Continued GEOMETRIC SKILLS Basic Geometric Figure Concepts Methods of Proof and 23.2 Reasoning Lines, Angles, Planes Congruence

Parallelism in Lines 23.5 And Planes Inequalities 37.9

Similarity 30.4

Locus 40.6

PolarCoordateFunct, Relations & Graphs 30.7

Coordinate Geometry

Constructions 31.4

Mensuration, Plane

Mensuration, Solid

TRIGONOMETRIC SKILLS

Angles and Coordinates Trigonometric Functions Trig. Functions by Arc Length Method Trig. Functions by Unity Method

Trigonometric Tables

Slide Rule, Trig. Functions Right Triangles, Trig. Solution of Periodic Functions 50.2 Fig. 3.--Continued TRIGONOMETRIC SKILLS

Plane Vectors 46.1 Oblique Triangles, Solution of Analytical Trig., Formulas 57.1 Analytical Trig., 60.9 Identities Analytical Trig., 59.7 Eivationa Graphs of Trig. 55.4 Functions Complex Numbers and 72.4 Position Vectors 11111111111E21*5 DeMoivre's Theorem 14.9 83..8

CALCULUS SKILLS

Variables, Functions And Relationships Limits -27.9 70.0 Differentiation of 11111111111124.0 74.5 Poloynominal !Unction Basic Theorems of Differentiation Integration 26.4 72.1

Integration Formulas 2.5 76.8 Trancendental 90.5 Functions Methods of 2.8 76.8 Integration Parametric Equations 89.1

Pb.'ar Coordinates

integration Formulas 2.8 76.8

Infinite Series 88.4

Hyperbolic Functions 91.3

17 Fig. 3. - -Continued CALCULUS SKILLS

Slide Rule, Hyperbolic Functions Partial 90.6 Differentiation 1111-649.7 928 Multiple Integrals 7.2 1

Differential 90.6 Equations Vector Analysis 111.01.9.8 88.7

ADDITIONAL SKILLS

Analog and Digital Computers

Operation of 4 21.2 Calculating Machines

Probability 55.3 Applied Statistics 57.6 And Dynamics ti 70.7 Logic Systems 111111111W-27.2

Boolean Algebra

Vector Analysis

Curve Fitting 62.8

Set Theory 11111-042.8 86.1 Root Approximation 1111111b.6.2 82.6 And Identification

Mathematical Fundamentals by Subjects proportion. Logarithms, indexing, gears, gear ratio, screw threads, spirals,complex gear ratio, and lead The frequency rating of topics as essential,desir- screws were all rated law. able, and not needed (for the subjectcategories of In algebra the fundamental operations,ratio, pro arithmetic,algebra,geometry,trigonometry, and portion, variation, fractions, and equations allrated calculus as summarized in Figure 3)shows which high. The advanced topics were not deemed essential. fundamentals were deemed essential. It also gives an indication of topics that were not important tothe Basic figures, lines and angles were considered mathematical qualifications of technicians. important in geometry, but other topics fell only into In arithmetic, those skills thm. were rated most the desirable range. important were: fundamental operations,fractions, Right triangletrigonometry,includingbasic standard notation, significant figures, approximation, angles, functions, and tables were rated essential. estimation, conversions, measures, tables, ratio,and Analytical trigonometry was not considered essential.

18 CHAPTER III Summary, Conclusions, Observations and Recommendations

The summary presents the procedural highlights dated 36 previously named firms by designating con- of the investigation. These are followed, in answer tacts at the corporation level. Some firms were deleted to the objectives stated on page one, by ,the conclu- as not suitable for the inquiry. Finally, 79 firms were sions of the study. In addition, the researcher assumes selected which was 44.8 percent of those that qualified. the prerogative of making some additional clearly Of this number, 71 (89.9 percent) were interviewed. related observations. Also, he presents his predictions of future mathematical needs of technicians in Michi- Interviews gan, and finally a list of recommended mathematical Management personnel, as directed by the top fundamentals. administrative officer of the company, were inter- viewed. In some of the smaller companies, one or two Summary of the Study persons participated, but in the larger firms the inter- Technician Needs view was in the form of a series of contacts with vari- ous persons, or was handled by a conference style Michigan, like other industrial states in America, interview. A tape recorder was used, together with a has a labor shortage and a labor surplus. The surplus set of prepared questions, to determine management is among the unskilled workers and the shortage is views of the mathematical needs of their technicians. among the skilled workers, especially in the technician The interview served also to identify the types of classification. With increased automation and techno- technicians that were employed by each company. logical advance in industry, the technician is one of The use of a