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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.

It is assumed that the geometry course refers to one that is commonly taught in the tenth school year. It is traditional Euclidean synthetic geometry, of 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.

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 to be noted that they (The Royal Society) have freely admitted Men of different Religions, Countries, and Professions of Life. This they were obliged to do or else they would come far short of the Largess of their own Declaration. For they openly profess, not to lay the Foundation as English, Scotch, Irish, Popish, or Protestant Philosophy but a Philosophy of Mankind.1

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.

The “breakthrough problem” in our school mathematics program is of national concern. As such, it becomes the concern of all teachers of mathematics.

The following method is suggested for the treatment of the quadratic equation in the teaching of algebra. The method makes use of factoring and substitution in their simplest form. If the quadratic is incomplete it can be factored as the difference of two squares, hence given the equation.

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.

“To be more specific, I am opposed to the introduction into the secondary school of the calculus in any form, of projective geometry, of analytic geometry as such, or of any special stress on the concept of function.” Thus Ralph Dennison Beetle1 of Dartmouth College rejects in part the recommendations of the National Committee, which say that “The one great idea which is best adapted to unify the course is that of functional relation” and which also include calculus in every plan of arrangement of material for the twelfth year. Having taught analytics and calculus as such, for the past four years to twelfth year students and having stressed the function concept in the eleventh and twelfth years, I am in direct opposition to Professor Beetle's opinion. In this article are presented some of the advantages that have resulted from this teaching which seem to make it desirable as a required part of the advanced mathematics in the secondary school.

Editor's Note.—In April 1966, at a meeting of ministers of education in the Arab states, held at Tripoli, a resolution was adopted requesting an updating of instruction in school subjects—particularly mathematics, pure and applied science, and foreign languages. In November 1966, the General Conference of UNESCO invited member states to undertake a major program for improvement of science and mathematics instruction and selected the Arab states as an initial place to start because of their April resolution. Mathematics was selected as the initial subject primarily because, worldwide, the reforms in education during the last fifteen years began with mathematics.

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.

I thought of speaking about Science, Mathematics and the Imagination, but I have changed my mind. I fancy enough has been said on this kind of subject to last us for some time. I don't believe that many sentient educated people nowadays doubt that creative mathematics and science call for as high qualities of imagination as any human activity: if they do, they have no excuse. And I believe people are beginning to realise that creative science doesn't work as scientists used to pretend it did. Medawar's writings1 and James Watson's recent book2 have taught the simple lesson that, as a rule, scientists know the answer before they prove it. This is called the hypothetico-deductive method. All this is very interesting, but it is becoming part of the intellectual climate. So I have decided to say something on a more practical topic, where there are some decisions we may have to make—some of them quite soon. I am calling this topic Elitism and Excellence.

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

Foreword: In proposing a radical departure from an existing program in mathematical education, one is always open to justifiable criticism from those who would maintain the status quo or those who would proceed otherwise in reform. One who seeks a change also has to some extent a bias for his point of view, which may prejudice the proposal. The following thesis, supported in part by experiments now under way, although presented strongly, is not advocated as THE program but is presented for consideration and dialogue aiming toward educational development.

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

When Perry made his now famous address to the British Society in 1901, he advocated a change in teaching, away from the pure abstract approach, and in its place, the use of scaled drawings, graphs, models, physical apparatus, and practical applications, so that pupils would achieve meaning or “mental tools as easy to use as their legs or arms; enabling them to go on with their education (developing of their souls and brains) throughout their lives, utilizing for this purpose all their experience.”

An extension of the method of defining a plane angle analytically to the definitions of a dihedral angle and the steradian.

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.

For a number of years past, conventional academic subjects have been subjected to critical analysis both by general educators, and by subject matter specialists.

An algebraic approach to some problems untouched by classical Euclidean geometry.

The task of identifying giftedness is not an easy one. In December 1940, a two- day conference and workshop on education for the gifted was held at Teachers College in honor of the great work done by Leta Hollingworth. One section of outstanding teachers and educational research workers devoted itself entirely to the task of identifying the gifted child. The conclusion reached was: “At the present time we have practically no adequate instrument for identifying the gifted.”1 In The Gifted Child2 edited by Paul Witty, we read, “Present means of identifying and guiding the gifted leaves much to be desired,” and the rest of the brief chapter gives adequate support to this stand both in its meagerness of discussion and the problems for investigation that are raised. Even the latest book on Educating Gifted Children3 a report on the Hunter College Elementary School Program by Gertrude Hildreth and others, takes the same point of view regarding our ability to detect the gifted at an early age. Formal tests seem to be the one criteria that most people rely upon.

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.

Despite the ready accessibility of elementary proofs1 of the impossibility of the Euclidean trisection of an angle, every year produces its crop of would-be solvers of the problem.

The Committee on Nominations and Elections presents its slate of nominees for offices to be filled in the 1960 election. The term of office for the president and the two vice-presidents is two years. Three directors are to be elected for terms of three years.

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.