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The questions of problem-solving and the nature of trial and error, and the insight in arriving at meaningful solutions to thought problems, have been investigated by a rather large number of psychologists. These researchers have been interested in the scientific aspects of the process, in finding out what it is that goes on in the mind or the organism as they solve problems. They are busy with the logical structure and explanation of thought, analysis, insight, etc. They have not, however, been concerned with how to initiate or to direct mental processes that lead to successful solutions. This latter is the task of the educator and the teacher, namely, to apply to the teaching process the findings of psychologists so that they may develop a greater problem-solving ability on the part of their students.

Fehr covers a wide range of topics in this article by necessity, because of his broadly conceived view of arithmetic instruction. He reminds us that arithmetic instruction must be determined by a set of several interrelated concerns: societal and children's needs, the structure of the discipline, and a knowledge of how children learn. Perhaps more than anything, though, I consider this article a classic because of Fehr's discussion of problem solving. To Fehr, problem solving is the ultimate goal of arithmetic instruction—a position that has a very familiar, contemporary ring. However, Fehr's views about how to develop problem-solving ability. which were an outgrowth of his commitment to field theories of learning, are less familiar and largely ignored. Since some recent studies tend to support Fehr's views, they warrant the attention of classroom teachers and researchers alike. By reprinting this article from April 1955. it is hoped that another perspective will be injected into the debate on how to teach problem solving. (The reader is referred to another classic by Fehr, “Theories of Learning Related to the Field of Mathematics,” in The Leaming of Mathematics: Its Theory and Practice, the Twenty-first Yearbook of the NCTM.)

It is assumed that the geometey course refers to one that is commonly taught in the tenth school year. It is traditional Euclidean synthetic geometry, 2- and 3-space, modified by an introduction of ruler and protractor axioms into the usual synthetic axioms. A unit of coordinate geometry of the plane is usually appended. It is a course that is reflected in textbooks prepared by the School Mathematics Study Group and in most commercial textbooks.

Not long after American teachers began the current reconstruction of the secondary mathematics curriculum, British mathematicians and educators, also recognizing the need for rejuvenation, instituted several projects for reform of an out-of-date curriculum. Among these, the programs of the School Mathematics Project and the Midlands Mathematical Experiment are most frequently cited.

Ever since the results of the International Achievement Tests in Mathematics were announced, much interest has been expressed in the outstanding performance made by the Japanese students.1 The test results displayed an accomplishment that at age thirteen, and overall, was far su perior to the other countries involved. Many analyses have been made of the subject matter involved in the questions, of the type of questions, of the teachers of the students, and of the curricula of each of the countries, to either sustain or disprove popularly expressed superiority of Japanese mathematics education. This article merely reviews the existing structure of Japanese education and points to some unique features of Japanese culture that may offer a partial explanation. It is not intended in any sense to suggest that the Japanese program or its culture— which is evidently satisfying to their own people—is one that would be satisfactory for others. It is merely one of many cultures which we should corne to under stand as the world grows closer knit in its forward progress.

The program below is the official program for all seventh grade students in Belgium, beginning September, 1968.

In Japan, education is compulsory for all children for the first nine school years, of which the first six years are elementary, and the last three years are the lower secondary school. Admission to the subsequent three-year upper secondary school is by examination, and usually 30 to 40 percent of the ninth-year students pass the examination and enter the upper secondary school. The branches of the upper secondary school are liberal arts, vocational training, and music and fine arts, all of which prepare students for university or higher study. This report is concerned with the mathematics curriculum at the lower secondary and the senior high school levels.

In the educational literature much information exists on the teaching of fractions and operations on fractions. This literature includes an abundance of suggestions for developing meanings that may be assigned to a fraction in elementary school instruction. However, except in mathematical treatises, there is not an abundance of information for teachers intended for challenging a deeper understanding of fractions. In this article there is presented one way of conceiving of fractions as mathematical entities with the purpose of broadening the teacher's knowledge of this aspect of arithmetic.

The concept of need pervades all modern theory with regard to learning. Parents and teachers are asked to discover and help the child satisfy his needs. The child's needs are to be at the center of all educational programs. However, there has been little or no suggestion that the child must also sense the need of understanding his parents and teachers and of their guidance and counsel. The school has as its primary task not only to serve the child to satisfy his immediate needs but also to equip him for service to society in his later life. Hence the child has need for that elementary mathematics which is demanded of all people in our free democratic society. Children will recognize some of these needs in their own daily experience, but other needs must be shown the pupils through experience supplied by the teacher.

IN THE early 1950s the mathema ticians at the university level began justi fiably to criticize the mathematical prep aration of high school students for university study. Partly because of this criticism, a number of reform movements arose, of which those of the University of Illinois Committee on School Mathe matics, the Commission on Mathematics of the College Entrance Examination Board, and the School Mathematics Study Group were most notable. The activities of these reform groups brought about a great improvement of the mathematics curriculum, though maintaining the tradi tional separation of mathematics into yearly units of study, especially for algebra and geometry. Included in these programs were quite a few topics not found in the pre-1950 era, such as the structure of the several number systems, the early introduction of the function con cept, and a more rigorous presentation of Euclidean synthetic geometry. Likewise, the notions of sets and set operations were used to the advantage of better under standing. In the late 1950s the reform movement spread to Europe. A two-week interna tional seminar held in Royaumont, France, and attended by mathematicians and edu cators from Europe and the United States was organized by the Organization for Economic Cooperation and Development (OECD). The report of this seminar gave direction to the needed reform in sec ondary school mathematics in terms of new topics to be added, omissions of obsolete topics in algebra and geometry, and the updating of fundamental concepts (OECD 1960). However, the report did not produce a new program for school study. Subsequently, all European coun tries embarked on the reconstruction of new mathematics syllabi that, along with further OECD seminars, clarified the manner in which secondary school mathe matics could be presented so as to bring out its contemporary nature as a unified discipline (OECD 1961; OECD 1964). Further support of this goal was given by a group of distinguished American mathe maticians who convened in 1963 at Cam bridge, Massachusetts (Cambridge Con ference on School Mathematics 1963). All the foregoing activities indicated the manner by which experimental teach ing?making use of mathematicians, edu cators, and classroom teachers in an op erational research endeavor?could bring forth a more unified, more efficient, and more meaningful program in mathe matical education for university-bound students. A proposal to develop such a program was made to both the United States Office of Education and the Na tional Science Foundation for financial support. Both of these agencies acted fav

If there is one phrase today that is prominent in the literature and conversation of all persons, professional and lay, and yet very nebulous, ambiguous, and meaningless in the minds of most of these persons, it is “modern math” or “the new math.” Now Modern Mathematics, as exemplified in such topics as Modern Algebra, Linear Algebra, Point-Set Topology, Algebraic Topology, Finite Mathematical Systems, Set Theory, and the like, is an exceedingly abstract, logical, axiomatic, well-structured system of knowledge. There is no nonsense in these branches of mathematics. They are pursued by students majoring in mathematics and science at the upper university and graduate levels of study. They have no place—no place—in elementary school mathematics. To do any of this mathematics in the elementary school would be more than nonsense—it would be insanity.

To Give an entire account of the teacher preparation in all countries would indeed be a lengthy process, the writing of which would require several volumes.

IN MATHEMATICS, as in other areas of knowledge, shifts of emphasis and of content take place in instruction from one generation to the next. This is as it should be. Discoveries are being made in subject matter and in the way human learning takes place. During the last quarter of a century sufficient changes in mathematics knowledge, in mathematical procedures and applications, and in the explanation of human behavior have occurred to warrant a reappraisal of our program in mathematics education. This evaluation can serve as a guide to reorientation of mathematics instruction in the schools.

For a number of years, the Association of Professors of Public Instruction (APMEP) has studied the problems posed by the evolution of the sciences, especially that aspect of education which concerns instruction in mathematics. Through the auspices of its Commission on Research and Reform, which has taken the initiative to bring things together, contacts have been made with professors of other disciplines, with researchers, with psychologists, and with engineers. The action of the APMEP is not unknown to the membership of the Ministerial Commission for Mathematics Instruction, headed by Professor Lichnerowicz, whose first report, published in March 1967, marked an important change in subsequent procedures.

Although we now often use the term “middle school,” the educational tenets described in this article remain viable today. Students from the ages of 10 to 15 are prime for meaningful learning. The notion of a general mathematics curriculum containing substantial review of earlier concepts is not acceptable; significant mathematics and problem-solving techniques need to be taught. Just as it was fifty years ago, the role of the classroom teacher is significant in providing motivation and approval.

The statement that all men are created equal has all too often been interpreted to mean that there is only a sameness to humanity, and hence that all men are to have the same of everything in life, the same worldly goods, the same schooling, the same recreation, the same rights and privileges., the same mathematical education. Of course we know this is nonsense, recognizing that all of us differ in endowed talents, in degrees of performance, and in the types of instruction and schooling we should obtain. We must always remember that the essence of democracy is difference, not sameness, and that our schools must provide for this difference. Yet to preserve our democracy we must share a common heritage, a sameness that unites us as one nation, and we must likewise provide for this sameness in our schools. It is in this light that a mathematics program must be devised for the oncoming generation.

“What is a set?” It is generally recognized that the first step in the teaching of arithmetic in the elementary school is to develop an understanding of the counting or whole numbers, that is, the numbers 0, 1, 2, 3, and soon. At the university level, there are two prevalent ways of developing the theory of these numbers. The one is to assume the numbers exist as a set, induce on them five axioms, referred to as Peano's axioms, and then by pure logic develop all the properties of and operations on these numbers. In one sense this corresponds to the rote method of teaching in the elementary school where the children are taught to recite the names of these numbers in a given order, and then taught how to use them, that is, to manipulate the numerals naming these numbers according to certain rules and procedures.

There are persons who do not like to teach who are conducting classes in our schools. There are persons who do not like mathematics who have been assigned to handle algebra classes and to instruct in arithmetic. There are men and women who do not understand mathematics yet they are engaged by boards of education to instruct in the subject. There are people with no sympathy for, nor understanding of children's desires and ambitions who daily browbeat their assigned pupils into fearful obedience through routine lessons. There are pedants who asperse educational theory and methods of learning, seeking only to perpetuate in their pupils the memorized meaningless manipulation of symbols that constitutes their entire stock of knowledge. They are blind to the world about them. These are not the teachers of mathematics. But we are teachers of mathematics if we have the following attributes.

Like a bolt out of the blue, education in America has suddenly become alive and exciting with new thinking concerning the intellectual purposes and subject matter of formal education. It is the intention of the writer to review some of the reasons for this activity, especially with regard to mathematical education; to briefly survey what we mean by the adjectives “new” or “modern”; and, finally, to present a picture of the content, purposes, and trends of emerging mathematical education.

With the great surge in mathematics, both pure and applied, the scientific and industrial world has called for a reform, both in our school mathematics curriculum and in the training of teachers to teach a more modern program. Just how much mathematical and how much professional education training a teacher needs is a matter of conflicting philosophies today, between those who believe in a body of knowledge called professional education and those who constitute the set of pure scientists. Since schools and the purposes of school education in America are quite different from those in most European countries, a critical examination of programs in other countries may give some perspective by which some of our conflicts may be resolved.

Teachers of mathematics are by definition responsible for teaching mathematics. Good mathematics teachers are also interested in the general education of their students.

Publisher Summary This chapter discusses the concepts of significance in instruction in the elementary school. The approach is not a psychological one, but rather one of a substantive and practical nature. A mathematical concept, even the most elementary or so-called basic one, is a very complex entity. Similar to the structure of an atom that in its nuclear description becomes a more and more complex structure, the apparently simple idea of cardinal number becomes a multiplicity of ideas on investigation. A general concept of number is attained only by very few persons after many years of mathematical study. The chapter reviews some basic concepts of elementary school mathematics such as set; mapping; relation (equivalence, order); cardinal number; counting; numeration; operation, algorism; and classification of numbers; fraction; rational number (positive or zero); measure; ratio, proportion, and percent; and space.

More and more the only services which human beings can offer to society and which society will need will be intellectual. Just as in the last 100 years mechanical skill superseded and to a great extent outmoded manual labor, in the next few years the automatic machine will supersede and to a great extent outmode mechanical skill (not entirely but mechanical skill will be cheap, comparatively to brain power). The age of automatons, for their development, construction, and maintenance, demands from society greater brain power.

A column of unofficial comment

A column of unofficial comment

THE NATIONAL COUNCIL of Teachers of Mathematics is deeply concerned with and interested in the teaching of arithmetic. It recognized that arithmetic is the basis of all succeeding study of mathematics. The Council has made sure that problems on teaching arithmetic will be given adequate attention by including on its Board of Directors a vice president at the elementary school level. Usually several other members of the Board of Directors are also directly involved in the improvement of instruction in arithmetic. There now exists a definite program of service to the instruction in elementary school mathematics. This program consists of three major areas, namely, publications, meetings, and committees.