general mathematics: the future as the past?

General Mathematics: The Future as the Past?Donald Paige

University Junior Hi^h School, Indiana. University, Bloomin^tonf Indiana

WHAT IS GENERAL MATHEMATICS?"General mathematics has one thing in common with the typical

virus�if we could identify or define it, we could do something aboutit."12 This opinion of general mathematics as given by Lentz in 1954is still valid today. It will be the purpose of this paper to review themeanderings of general mathematics from its inception during theutilitarian movement of the 1910’s to the present. Also, the investi-gator will suggest what he feels should be the future of general math-ematics.

For the remainder of this paper the term "general mathematics"will represent all courses in high-school mathematics which aredesigned for remedial purposes, vocational purposes, and consumerpurposes. Not included in the term "general mathematics" arealgebra, geometry, and other courses designed for advanced mathe-matical use. The term "sequential mathematics" will be used inreference to courses not included in the term "general mathematics."By high-school mathematics the writer means the mathematicsfound in grades nine through twelve even if the ninth grade class islocated in a junior high school.

HISTORY OF GENERAL MATHEMATICS

General mathematics was introduced into the high-school cur-riculum during the first two decades of the twentieth century. Theutilitarian motive was the vogue in public education at that time,and the existence of parallel curricula, college preparatory andvocational, called forth a new mathematics course which spawnedthe multitude of courses presently known as general mathematics.

General courses such as Mathematics for Industry, ConsumerMathematics, and Basic Mathematics and specific courses such asSurveying, Mathematics for Investment, and Shop Mathematics weresoon offered in various schools in an effort to have a mathematicscourse available for every student regardless of his or her vocationaldirection. These general mathematics courses were, in many cases,designed to fit the needs of specific companies, or specific vocations.Thus the utilitarian motive which started general mathematicscourses was augmented by the motive of mass education, and thecourse offerings in this field continued to grow with no apparentdirection or control. At this time the publication of a new text wouldstart a new course.

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196 School Science and Mathematics

One of the first directive forces on general mathematics was the23rd Yearbook of the National Society for the Study of Education, PartII, half of which is devoted to the vocational needs of students. Thisreport published in 1923 was the first of three early, importantstudies which tended to unify the various offerings into the fieldnow known as general mathematics. Prior to this time, indeed eventoday, some of the courses were offered at the high-school level bydepartments other than mathematics departments.The study by Charters published in 1926 further augmented the

unification of courses.7 Until this time courses were being offered forspecific vocations, and the mathematical content was duplicated inmany of them. Charters’ study consisted of a job analysis whichfound what traits or skills were needed for certain vocations. Thebasic mathematical skills were found to be similar in various voca-tions, which allowed educators to combine multiple general-mathe-matics offerings into a few courses with emphasis on various applica-tions along with the skills involved.The multiple offerings persisted, however; the urbanization move-

ment of society, technological growth of science, industrial growth,and the growth of various services probably were responsible. Skillsin mathematics were probably thought to be more varied since newapplications were constantly being discovered by the growing, com-plex society of this country.A study conducted by Bobbitt in 1925 augmented the growth of

general mathematics.3 His survey of popular-science magazines andbooks was intended to discover the mathematical skills needed bythe average citizen. To this writer it seems that he probably dis-covered what skills were known by the then-current writers ofmagazine articles and books on science. However, the skills reportedby Bobbitt were probably the basis for many texts published duringthe decade which followed.

Surviving, indeed growing, during the depression years generalmathematics was again brought into focus, this time by the NationalCouncil of Teachers of Mathematics. In his article ^Arithmetic inGeneral Education/’ which appeared in the National Council’s 16thyearbook, Benz implied that arithmetic must be kept in the highschool curriculum, but more use must be given to vocational andsocial problems.2 Persistent then, as they have been throughout thehistory of general mathematics, were educators who insisted thatonly college preparatory mathematics be taught in high school.Presumably Benz’s article was directed toward them.The 17th Yearbook of the National Council of Teachers of Math-

matics, the next published after Benz^s article, was devoted to every-day applications of mathematical skills. This publication seems to

General Mathematics 197

proclaim that the National Council of Teachers of Mathematicsrecognized the value of general mathematics even though most ofits members felt insulted if they were asked to teach a course in thefield.The general mathematics question was further confounded when

the Commission on Post-War Plans published their first report in1944.8 They suggested three general types of courses be offered inhigh school mathematics; first, sequential mathematics for the col-lege-bound student; second, related mathematics which could betterbe called vocational mathematics or industrial mathematics; andthird, social mathematics which could better be called consumermathematics.The Commission on Post-War Plans said what should be accom-

plished, but this by no means ended the differences in generalmathematics. More has been written about it in the past decadethan in the four previous decades. Also, probably more differentattacks and defenses have been made against, and for, generalmathematics than any other area in the high-school curriculum.Whitman represented a large group who insist that general math-

ematics must not repeat earlier work when he stated:It would seem that if the student hasn’t mastered the fundamentals after

studying arithmetic through the eighth grade, continued repetitious drill wouldserve only to set up a psychological block.18

Dyer, Kalin, and Lord felt that the course does not cover thematerial for which it is designed when they wrote:

This course has been conceived in different ways. Some people have thought itshould develop an elementary understanding of algebra and geometry; othershave thought it should deal with those topics the ordinary person needs to knowin order to conduct his daily affairs. In practice it has usually become a final at-tempt to pound home the basic operations of arithmetic, an experience whichteachers have not found exciting. Students, too, look down their noses at generalmathematics, but take it to avoid algebra.19

In contrast Wilson said, ^In my opinion, courses in consumermathematics do include many socially useful topics, but these topicsdo not involve enough real mathematics. . . . We are living in aperiod of such rapid change that we cannot possibly plan an educa-tional program on the basis of social utility," which seems to implythat we should teach basic mathematics and not social applications.19The investigator thinks that basic mathematics must be stressed,but in addition feels that social applications should be employedinstread of drill on abstract mathematical problems. Whatever theopinions of writers and course designers may be there exist manyarticles listing steps to be followed in the selection of subject matterto include in the course. One of these is the article by Irvin whichstates:


Varied criteria should be used in the selection of subject matter content for thecourses in each "track": (a) current, successful practices, (b) the nature of childgrowth and development, (c) the recommendations of specialists, and (d)the de-mands of society, both at present and in the near future.10

The content of general mathematics is one question; the placementof general mathematics is another. Brown felt, however, that this isno problem.5 He said that general mathematics can be offered atany grade level, seven through twelve. The present writer, however,limits the use of ^general mathematics" to mean courses offered ingrades nine through twelve, as pointed out earlier.

Schult implied that we must teach general mathematics in theninth grade to those students who are not in algebra.17 At least thisis the only way that non-algebra students can master the list ofessentials which she felt all ninth graders should know.On the other hand a survey of schools in Nebraska showed that

students do not gain the essentials they need in either algebra orgeneral mathematics. In this study Beckmann concluded:The most general and obvious conclusion that can be drawn from the study is

that the vast majority of students are leaving the ninth grade with a woefullyinadequate understanding and command of the essentials for functional com-petence in the area of mathematics.1

This discussion as to placement leads eventually to the conclusiondrawn by Kinney when he stated:

Experience reveals that one course in general mathematics cannot achieve this.Even if there were time for all the problems that should be dealt with, the ninthgrade pupil is not sensitive to the significance of many of those that will appeal tohim later on as a senior.n

The investigator feels that more than one year of general mthe-matics is required if we are to give our students the mathematicalskills and applications needed in their future.The solution of placement and content in general mathematics

still leaves one major problem, acceptability. As stated earlier, somemathematics teachers are insulted if they are asked to teach evenone section of general mathematics. In fact, some teachers have beenknown to resign when they were assigned general mathematicscourses. The problem of acceptability leads to stigmatized students,which in turn creates more problems than our guidance departmentswant to handle. To avoid the problem of stigmatized students generalmathematics must be recognized as an accpetable course. Lentz putsit bluntly when he says, ^General mathematics will be as respectableas the administration and staff make it."12

Disregarding the problems which have faced general mathematicsin its past history and the problems facing it today, educators arechallenged by Cannon^s Statement, "High schools which haven’tprovided for slow learners should plan such a course immediately,"6


and by the further fact that ^33.1 per cent of ninth graders enrolledin general mathematics during the 1956-57 academic year."15 If theeducational system in this country is to serve all people then anattack must be made on the diversity and incompetence which seemsto exist in the area of general mathematics and, indeed, has existedsince its beginning fifty years ago.

THE FUTURE or GENERAL MATHEMATICSIt seems obvious that some change is necessary in the field of

general mathematics. Brown pointed out two shortcomings when hestated:There are at least two fallacies in the status-quo curriculum approach. First,

industry might require a greater use of mathematics if the workers had greaterability in the subject. Second, when the present pupils are adults, a greater knowl-edge of mathematics may be demanded of semi-skilled workers than is now thecase.4

This and other statements must be taken into consideration in theformation of a new course of study for general mathematics.The investigator feels, as do many others, that two years or more,

should be required for students who are not enrolled in the sequentialmathematics courses. One of these courses should be offered duringthe ninth-grade year and the other during the last three years ofhigh school with preference for the senior year. Within a short timethis problem may be further confounded by a coming trend ineducation, i.e., required attendance until pupils are eighteen years ofage. This new trend would increase the need for more instruction ingeneral mathematics since the students it would keep in school arethose who would be directed into general mathematics courses andwould benefit most from them.The objectives for general mathematics are generally very complex

and wordy. The outcomes though, as mentioned earlier, are not sobroad or significant. Bulletin number 212 of the Indiana Depart-ment of Public Instruction lists eight objectives for general mathe-matics, but this is preceded by the suggestion that the major empha-sis is to be on basic arithmetic skills and understandings.13

This writer would like to limit the various profuse objectives ofgeneral mathematics to two simple statements: (1) Show your stu-dents the beauty of mathematics, and (2) teach your students themathematical competence needed for their future vocations andavocations.To accomplish these objectives the writer envisons two math-

ematics courses, one for each of the objectives listed above. For thefirst objective and the ninth-grade year of general mathematics thiswriter suggests an introductory course which would cover manytopics in mathematics.


This writer feels, as do many others mentioned earlier, that ninth-grade general mathematics should not be a repetition of the arith-metic work covered during the seventh and eighth grade years. Ifwe are not to have repetition we must introduce new topics into ourcourse of study, even topics which are new to some of the teacherspresently teaching general mathematics. Such topics as logic, simplelinear programming, probability and statistics, simple matrix opera-tions, and others should be included in addition to the concepts fromthe traditional algebra and geometry courses. Justification for theinclusion of these topics could follow the typical pattern; that is,these topics are needed for success in later work or they are neededfor success in life. Better justification, at least to this writer, is thefact that most students enjoy these topics. If students are to enjoythe beauty of mathematics they surely must enjoy some of the classwork with mathematical ideas and concepts. If students enjoy codingand decoding messages with the use of matrix multiplication, if theyenjoy logic as explained by Venn diagrams, if they enjoy industrialproblems as solved by linear programming, and if they enjoy theprobability side of games of chance, then these topics should be in-cluded in the course of study.A ninth-grade general-mathematics course must create interest

before the students will make any effort to learn the skills they dis-liked in earlier years. Although learning arithmetic is not the objec-tive of a course of this type, it would be an outcome since competencein basic arithmetic computations is required for successful work insome of the topics mentioned earlier.

For the second year of general mathematics and for the secondobjective the writer suggests the creation of a new course, SeniorGeneral Mathematics, based on the mathematical skills and conceptsneeded in various vocations. These needed abilities are not readilyavailable and a good regional survey of the vocations should be con-ducted to discover the mathematical concepts involved. Whateverthe results of a survey, the senior course in general mathematicsshould be designed to cover basic concepts in all areas rather than,as is now the case, offering various courses with specific problemsfrom specific vocations.

SOME QUESTIONSThe pendulum of interest has swung far from general mathematics

in the last decade. Modern mathematics and experimental projectshave given little consideration to the area. Are the varied offeringsof general mathematics too complex for consideration? Is the pasthistory of general mathematics so confusing that its real purpose islost? Can we afford the time and money invested in college prepara-


tory programs while no improvement is forthcoming in generalmathematics? Is general mathematics always to be the second-ratearea for the second-rate students?

BIBLIOGRAPHY1. BECKMANN, MILTON W., "How Mathematically Literate is the Typical

Ninth Grader after Having Completed Either General Mathematics orAlgebra?," SCHOOL SCIENCE AND MATHEMATICS, 52: 449-455, June, 1952.

2. BENZ, H. E., "Arithmetic in General Education," in 16th Yearbook of theNational Council of Teachers of Mathematics, Columbia University, NewYork, 1941, pp. 119-139.

3. BOBBITT, FRANKLIN, Curriculum Investigations, The University of Chicago,Chicago, 1926, 202 pp.

4. BROWN, KENNETH E., Analysis of Research in the Teaching of Mathematics1955 and 1956, Office of Education, bul. no. 4, U. S. Department of Health,Education, and Welfare, Washington, D. C., 1958, 71 pp.

5. BROWN, KENNETH E., "Status of Mathematics Education in Public Second-ary Schools," The Bulletin of the National Association of Secondary SchoolPrincipals, 38: 54-62, May, 1954.

6. CANNON, NORVAL L., "Slow Learners May Need ^Personalized^ Mathema-tics," School and Community, 49: 11, January, 1963.

7. CHARTERS, W. W., "The Foundations and Techniques of Curriculum Con-struction," in 26th Yearbook of the National Society for the Study of Education,Public School Publishing Company, Bloomington, Illinois, 1926, pp. 365-379.

8. Commission on Post-War Plans, "The First Report of the Commission onPost-War Plans, "The Mathematics Teacher, 37: 226-232, May 1944.

9. DYER, HENRY S., KALIN, ROBERT, AND LORD, FREDERIC, Problems inMathematical Education, Educational Testing Service, Princeton, 1956, 202pp.

10. IRVIN, LEE, "The Organization of Instruction in Arithmetic and BasicMathematics in Selected Secondary Schools," The Mathematics Teacher,46: 235-240, April, 1953.

11. KINNEY, LUCIEN, B., "Mathematics," in The High School Curriculum,edited by Harl R. Douglas, The Ronald Press Company, New York, 1956.

12. LENTZ, DONALD W., "The High School Principal Looks at the MathematicsProgram," The Bulletin of the National Association of Secondary School Prin-cipals, 38: 40-47, May, 1954.

13. Mathematics for Secondary Schools {A Guide to Minimum Essentials), bul. no.212, State of Indiana Department of Public Instruction, 1963, 54 pp.

14. National Council of Teachers of Mathematics, l7th Yearbook, Bureau ofPublications, Columbia University, New York, 1942, 291 pp.

15. National Council of Teachers of Mathematics, Some Pertinent Facts Aboutthe Current Status of High School Mathematics and Science, (mim. report),1958, 3 pp.

16. National Society for the Study of Education, 23rd Yearbook: Part II, PublicSchool Publishing Company, Bloomington, Illinois, 1924, 455 pp.

17. SCHULT, VERYL, "Guideposts in Curriculum Planning in Mathematics," TheBulletin of the National Association of Secondary School Principals, 38: 48-53,May, 1954. �

18. WHITMAN, SOL, "We Can Dress Up General Mathematics," SCHOOL SCIENCEAND MATHEMATICS, 52: 210-212, March, 1952.

19. WILSON, JACK D., "What Mathematics for the Terminal Student," TheMathematics Teacher, 53: 518-523, November, 1960.

general mathematics: the future as the past?

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