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Contemporary Goals for Mathematics Instruction

Donald R. WhitakerBall State University, Indiana

A major goal of education in a democratic society is to produce citi-zens capable of intelligent, independent thought. Educators at all levelscontinually strive to attain that goal and, toward that end, often organizeseries of goals within their respective disciplines. Historically, in the dis-cipline of mathematics, various individuals and groups have recom-mended such sets of goals (for example, see CEEB, 1959; CambridgeConference on School Mathematics, 1963; Buck, 1965; CBMS, 1975).Recent technological and societal developments suggest it may be time toconsider an up-dated list of goals for mathematics instruction. Thispaper examines what the author believes are major factors influencingmathematics instruction in the waning years of this century. It concludeswith a list of eight contemporary goals for mathematics instruction.

^Societal trends clearly dictate a movement toward moretechnology, not less/9

The determination of appropriate goals within any discipline is influ-enced not only by the inherent nature of the discipline itself, but also byexternal factors. For example, the so-called "basic skills" movement ineducation has caused a great deal of concern among mathematicseducators. This concern centers on the desire that skills deemed "basic"not be narrowly defined and has caused several professional organiza-tions to endorse a broadly-defined statement of the "essentials of educa-tion" (see NCTM Newsletter, March 1980).Another factor which has far-reaching implications for mathematics

instruction is the widespread availability of relatively inexpensive butsophisticated microprocessing equipment. Schools are finding that vari-ous computing units are now affordable. Many families have microcom-puters in their homes. Student interest in computing is high. And what ofthe nearly universal availability of hand-held calculators? Societal trendsclearly dictate a movement toward more technology, not less. Certainlyour technological advances suggest a re-thinking of our concept of"basic" or "essential" skills.

If we turn to the nature of the discipline itself, it is important to notethat mathematics is, above all, a human activity, something which one’

does, rather than something which one passively learns. Thus, the goals

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for its instruction should reflect a human involvement in that activity. Inhis essay, "The Aims of Education," written in 1912, Alfred NorthWhitehead captured a feeling for the involvement of the learner by ob-serving that children should experience the joy of discovery from the be-ginning of their education. Whitehead was also quick to point out that, ifa subject has any value, the student should be able to appreciate its im-portance immediately. Though he was writing about 70 years ago,Whitehead’s message has relevance for mathematics instruction today.The implications for learning seem clear; the mode of execution doesnot. But goals for instruction should merely locate direction, not chartthe exact course to be followed.Many would contend that mathematics is a human creation of reality,

born of a need to quantify our environment. Certainly students shouldcomprehend the importance of mathematics in helping us understandand master the physical, the economic, and the social worlds (CBMS,1975). The level at which a student comes to value and appreciate the sig-nificant applications of mathematics will vary with the individual, but aconcentrated effort to relate mathematics to the real world must be madethroughout a student’s formal education.One of the goals in a list advocated by R. C. Buck (1965) places em-

phasis on student awareness of the intuitive foundations of mathematics.Such awareness is salient, as many students have the mistaken notionthat the really important discoveries in mathematics somehow mysticallydescended upon the mathematicians responsible for them. Kline (1973)argues that the basic approach to all new subject matter at all levelsshould be intuitive. He comments further:

... He (the student) should be allowed to accept and use any facts that are so obviousto him that he does not realize he is using them. The capacity to appreciate rigor is afunction of the mathematical age of the student and not of the age of mathematics. Thisappreciation is acquired gradually and the student must have the same freedom to makeintuitive leaps that the greatest mathematicians had (p. 195).

The point at which a student realizes that the intuitive understandingsand agreed-upon conventions are not externally fixed will depend uponthe state of mathematical sophistication of the individual, but the impor-tance of this realization cannot be denied.

Certainly we would hope students realize early in their schooling thatthe very essence of mathematics is problem solving and that processesuseful for solving mathematical problems may be applied in a variety ofsettings and disciplines. Problem solving explorations in mathematicsalso point out the diverse nature of the subject and often suggest future

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avenues of research. And the ever-expanding body of research findingsin the mathematical sciences can withstand further bolstering.

This commentary doesn’t represent an exhaustive treatment of allfactors influencing mathematics instruction in schools. Those factors arefar too numerous to delineate here. The commentary is intended to setthe stage for an up-dated listing of goals for mathematics instruction.The eight goals given below are not directed toward any one level of in-struction, but rather cut across all levels. They reflect the author’s viewsresulting from over three decades devoted to learning and teachingmathematics at various levels of schooling.

". . . mathematics is an ever unfolding discipline 9 character-ized by structure, flexibility, rigor, induction, deduction,and simplicity ..."

Eight Contemporary Goals for Mathematics Instruction

1. Guide students in the active investigation of the world of mathematics around them,thereby instilling in them an awareness of the applications which mathematics has intheir lives.

2. Promote in students an understanding that mathematics is a human discipline builtupon intuitive notions and understandings of the real world.

3. Help students discover techniques of inquiry and develop the confidence to examine,question, solve, and validate mathematical problems, thereby formulating strategiesuseful for attacking the perplexities of life.

4. Aid students in mastering a core of "essential skills" peculiar to the discipline ofmathematics, but deemed necessary for life in a highly technological age.

5. Assist students in understanding that mathematics is the foundation for all scientificthought and that the interaction of mathematics with other disciplines helps to ex-plain the quantitative and qualitative aspects of our environment.

6. Identify and encourage mathematical creativity and help mathematically talentedstudents appreciate the beauty of mathematics and realize that mathematics is a dis-cipline worthy of study for its own sake.

7. Assist students in realizing that mathematics is an ever unfolding discipline, charac-terized by structure, flexibility, rigor, induction, deduction, and simplicity, withmany areas of the discipline open for future research.

8. Help students like mathematics, or at the very least, to enjoy the study of mathe-matics.

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The last of these goals probably deserves additional comment, since itis affective, rather than cognitive, in nature. The successful teaching ofmathematics, at whatever level, often is hindered by an expressed dislikeof the subject by students. Many students even experience what has beentermed "math anxiety" when attempting to deal with the subject. Theeighth contemporary goal merely suggests that we make a concentratedeffort to deal with this unfortunate situation. Kline (1973) recommendsthat we use the following criterion of success:

. . . When we reach the stage where fifty per cent of the high school graduates can hon-estly say that they like mathematics and appreciate its significance, then we shall haveattained a large measure of success in the teaching of mathematics (p. 204).

Surely this is a challenge to all who are interested in the teaching andlearning of mathematics.As a closing footnote it should be observed that many forces outside

the mathematics community will continue to have a strong impact on theproper goals for mathematics instruction. Teachers of mathematicsmaintain a constant effort to stay abreast of new ideas and developmentswithin the discipline, but they need the assistance of individuals frommany disciplines to be fully aware of the implications which social, scien-tific, technological, political, and economic factors have for the teachingof math. And the task of setting appropriate goals for mathematics in-struction, like the discipline itself, is ever-changing.

REFERENCES

1. BUCK, R. C. ^Goals for Mathematics Instruction." American Mathematical Monthly’,1965,72,949-956.

2. CAMBRIDGE CONFERENCE ON SCHOOL MATHEMATICS. Goals for School Mathematics.Boston: Houghton Mifflin Company, 1963.

3. CBMS, NATIONAL ADVISORY COMMITTEE ON MATHEMATICAL EDUCATION. Overview andAnalysis of School Mathematics Grades K-12. Washington, D.C.: CBMS, 1975.

4. CEEB, COMMISSION ON MATHEMATICS. Program for College Preparatory Mathematics.New York: CEEB. 1959.

5. KLINE, M. Why Johnny Can’t Add: The Failure of the New Math. New York: RandomHouse, 1973.

6. NATIONAL COUNCIL OF TEACHERS OF MATHEMATICS. NCTM Newsletter. March 1980.7. WHITEHEAD, A. N. The Aims a/Education and Other Essays. New York: The New

American Library, 1949.

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