
The Mathematics Curriculum for the High School of the Futureby Howard F. Fehr  1958 The mathematics curriculum of the future must meet the needs of mathematics, physical sciences, social sciences, engineering, technology, industrial management, and other areas of human endeavor as they are carried on in the second half of the twentieth century. If we are to proceed wisely in constructing a curriculum for the future we must examine first some defects of the present curriculum, second, the present mathematical needs of our society, and third the changing concepts and new developments in mathematics. Then we must temper this mathematical knowledge with our knowledge of the maturity and learning ability of the high school population. THE. present high school curriculum in mathematics is a traditional one. It is outmoded, oriented to nineteenth century mathematics and physics, and no longer provides for present or anticipated needs of high school students.1 The mathematics curriculum of the future must meet the needs of mathematics, physical sciences, social sciences, engineering, technology, industrial management, and other areas of human endeavor as they are carried on in the second half of the twentieth century. If we are to proceed wisely in constructing a curriculum for the future we must examine first some defects of the present curriculum, second, the present mathematical needs of our society, and third the changing concepts and new developments in mathematics. Then we must temper this mathematical knowledge with our knowledge of the maturity and learning ability of the high school population. THE TRADITIONAL PROGRAM Today the ninthyear mathematics program is commonly given to a study of elementary algebra. What is this study? The manipulation of symbols according to rules in a structureless system. There are no proofs, there is no system of axioms, there are no undefined terms; the instruction is mostly "how to do." Such terms as literal numbers, general number, unknown, and algebraic numbers are for the most part meaningless, or even confusing. A student learns how to solve equations, but not what an equation is, or what operations are allowable and why, or what the solution indicates. If he is asked to apply his knowledge of solution of equations to problems, the latter must be explained in great detail. That a student learned a set of operations, mostly meaningless to him, and that he can do little with them in any original setting has been shown again and again. In the tenth or eleventh year the study of this algebra is continued as intermediate algebra. How? By first reviewing the skill algebra of the ninth year in exactly the same way with more complex situations. Then the student learns "how" to handle exponents, logarithms, systems of equations, progressions, and the binomial expansion without any hint that proof in an axiomatic system is as essential in algebra as it is in dealing with geometric elements. If he is introduced to any other topic it is usually permutations, combinations, and probability. But these are taught in an antiquated manner, illadapted for application to modern problems in statistical processes. The function concept present in most texts is no longer generally accepted by mathematicians, and even when presented it contributes little to understanding this important aspect of mathematics. In either the tenth or the eleventh year the student also studies plane geometry, more correctly termed Euclidean synthetic plane geometry. What does he learn here? He gets an introduction into the physics of a plane through the use of definitions, undefined terms, and a set of assumptions. A yearlong chain of theorems follows that in most cases results in large quantities of mere memorization. At the end of the year the student may be a bit more clever at discovering deductions of "socalled" originals, but is he really a better mathematician with a real understanding of the axiomatic structure of mathematics? The answer must be NO! Twelfthgrade mathematics in the American high school is the grandest fiasco of all mathematical programs. A half year of solid geometry contributes nothing new to an understanding of mathematical structure. The geometry of the sphere, which could become a fine selfcontained unit of study, is buried under a heap of useless applications. To cap the climax we have a semester of trigonometry, which is usually concluded with two months of solution of oblique triangles by the use of logarithms. General angle, reduction formulas, identities, and equations as taught contribute little to an understanding of trigonometry that is of value for further work in mathematics. SOCIETY'S NEED FOR MATHEMATICS Now let us look at the role that mathematics is coming to play in our culture. I shall omit all reference to general education and the mathematical needs of all citizens as a part of their general education. It suffices to say that all citizens need a working knowledge of arithmetic, elementary algebra, and geometric relations and measures. All secondary school students should study mathematics until they have mastered these necessary facts, concepts, and skills. Here I would speak of the newer and more advanced mathematical needs of society. The degree to which mathematics is applied to the other sciences and to socalled nonscientific social activities has increased tremendously during the last few decades and is increasing continually. Mathematics has always contributed to the fields of physics, engineering, and technology. More recently mathematical methods have been applied to industrial planning, medicine, biochemistry, biophysics, and sociology. Even problems in philosophy and linguistics are being attacked through the use of mathematical logic. In all of this increased activity it is a curious fact that, although the first investigations were begun by mathematicians, on the whole it is not mathematical propaganda or advertising that has made the situation what it is. It resulted from a genuine demand on the part of workers in these fields who came to feel more and more helpless when they could not handle mathematical methods. The number and variety of mathematical disciplines have greatly increased in the last sixty years. New branches of knowledge based on mathematical methods have been created. Among these can be mentioned: Design of Experiments, Mathematical Population Theory, Theory of Risks, Symbolic Logic, Biomathematics, Factor Analysis, Quality Control, Mathematical Theory of Communication, Information Theory, Theory of Strategy and Games, Linear Programing, Periodogramanalysis and Time Series, and Statistical Decision Theory. While not all of these new theories have produced practical results commensurate with their mathematical structure, the judgment of workers in the field is that the mathematical approach has been on the whole beneficial to their particular domains. Mathematicians themselves are creating new branches of pure mathematics, much of it knowledge that did not exist sixty years ago. We may mention axiomatics, abstract algebra including the theory of groups, rings, fields, and vector spaces; combinatorial topology and algebraic topology; lattice theories; general theory of sets; theory of linear spaces; tensor calculus; and even metamathematics—a study about, not of, mathematics. I hasten to add that I know little of these fields of applied and pure mathematics except that they exist; that they have burst the existing compartments that house arithmetic, algebra, and geometry; and that in their very nature they make much of the classical treatment of high school mathematics obsolete. (Note that I said the treatment, and not the mathematics itself, becomes obsolete.) NEW POINTS OF VIEW In the past one hundred years the nature of mathematics as a subject has been substantially altered by the results of mathematical research and by its applications. Some of this research can have significant impact on the high school program. We can mention only a few of these newer areas of mathematical thought, however, without developing their content. In the field of algebra we mention first the concepts of group, ring, and field—all dealing with structural rather than manipulative aspects of algebra. None of these concepts is difficult to grasp. The concept of field enables us to study the structure of algebra rather than to merely manipulate algebraic expressions. This concept has given rise to studies of different algebras: linear associative, vector, multilinear, and so on. This aspect of algebra has reshaped the thinking and the type of research that are now being done at the frontiers of mathematical knowledge. An advanced treatise in mathematical analysis of the 1920's30's, always had a first chapter that contained a treatment of the real number system upon which, of course, the rest of the development rested. If you pick up treatises today, either in advanced algebra, analysis, or geometry, the first chapter is always a treatment of the theory of sets upon which the rest of the development rests. Set theory is the newest unifying concept to enter the field of mathematics. The ideas of sets in modified form can be applied to every portion of the present high school program. In the field of geometry, new points of view are emerging also. A hundred years ago nonEuclidean geometries became firmly established as worthy structures of mathematical study, thus freeing the subject from the absolute authority of Euclid. With the advent of these new geometries there arose a critical examination of the axiomatic foundations of Euclidean geometry. This study revealed subtle ideas in the subject that proved it to be too difficult a field for rigorous treatment at the high school level of study. The researchers also established other geometries called protective, affine, elliptic, and hyperbolic, and shattered forever the idea that there is only one kind of geometry. Finally, a new approach to the study of space, called Topology, has opened fields of investigation in which the very first findings are of importance to the present high school geometry program. THE CURRICULUM OF THE FUTURE Considering society's demands for mathematics and the modern developments in mathematics mentioned above, we can draw some inferences concerning the curriculum of high school mathematics of the future. We consider here both the junior and the senior high school. Seventh and Eighth Grades. The first two years of the junior high school will offer an intuitive and informal study of arithmetic, geometry, and some elements of algebra. More precisely, at the end of this informal study, a student should have a mastery of the four fundamental operations with whole numbers, common fractions, and decimal fractions. This includes skill in the operations at adult level (that is, adequate for ordinary life situations) and an understanding of the rationale of the computational processes. A placevalue system of numeration with special reference to our decimal system should be understood. Systems of numeration with other bases, particularly the binary system should be investigated. Pupils must be able to handle very large numbers (greater than 1,000,000) and very small numbers (less than one tenthousandth, 0.0001). In addition, a knowledge of square root and the ability to find approximate values of square roots of whole numbers by the process of division and averaging the divisor and quotient (Newton's method) are recommended. An understanding of the language of per cent (rate), percentage, and base is essential and in particular, the ability to find any one of these three designated numbers given the other two. The ability to treat with confidence per cents less than i and greater than 100 must be acquired. Applications of per cent to business practices, interest, discount, and budgets should be given only moderate treatment. The study of arithmetic must include the understanding of ratio as used in comparing sizes of quantities of like kind, in proportions, and in making scale drawings; also the ability to operate with and transform the several systems of measure, including the metric system of length, area, volume, and weight. The nature and use of an arithmetic mean are to be stressed. The work should be directed toward an informal study of algebra. The informal geometry must include the study of length of a line segment, perimeter of a polygon, and circumference of a circle, area of regions enclosed by polygons and circles, area of solids, volumes enclosed by solids, interior of angles (by degrees). In this work, the use of a ruler (both English and metric) and protractor is learned. The pupil should know the difference between the process of measuring and the measure of an entity and should develop the ability to apply measurement to practical situations. He uses measurement in drawing to scale and finding length indirectly. Further concepts of geometry that should be developed are those of parallel, perpendicular, intersecting, and oblique lines (in a plane and in space); acute, right, obtuse, complementary, supplementary, and vertical angles; scalene, isosceles, and equilateral triangles; right triangles and the Pythagorean relation; sum of the interior angles of a triangle; sides and interior angles of a regular polygon with six or fewer sides. The pupil develops skill in the use of instruments in constructing figures; he learns ideas of symmetry about a point and a line. Further ideas included in these two years of study are the use of a line segment and area to represent numbers, the reading and construction of bar graphs, line graphs, pictograms, circle graphs, and continuous line graphs, the meaning of scale, formulas for perimeters, areas, volumes and per cents—introduced as these concepts are studied—as mathematical models, the use of symbols in formulas as place holders for numbers arising in measurement, and simple expressions and sentences involving variables. Algebra. The study of algebra will consist largely of the same subject matter as hitherto. The difference will be in point of view, and this will be concerned principally with concepts, terminology, symbolism, and the introduction of a rather large segment of work on inequalities. The idea of onedimensional graphs, for example, indicating the set of points for which x<3 will be introduced. There will be a shift of emphasis from stress on mechanical manipulation to understanding of the fundamental ideas and basic laws. The study of the nature of number systems, the variable over the system, and the basic laws for addition and multiplication, namely the commutative, associative, and distributive laws, are focal points. The application of these laws in various algebraic systems, with emphasis on their generality, the meanings of conditional equations and inequalities, of their solution sets, and of equivalent equations and inequalities will receive as much attention as the mechanics for finding the solution sets. If this approach to algebra were merely something interesting but useless, something at an abstract level that could be learned by high school students, merely an extension of the algebra of the eighteenth century, or a game for pure mathematicians only, we could ignore it. But it is none of these. It is finding application in all the sciences pure and applied. The ideas are elementary and can give the "meaning" we have been looking for in the teaching of our algebra. It is new. It can be introduced at almost any time in the high school program. It gives to algebra a structural and unifying property. Hence, we cannot afford to ignore this approach as we reconstruct our high school program. Concepts do not remain static in mathematics. When Leibnitz in 1675 used the word function, he used it as concomitant change. Euler in 1770 said it was the relation between y and x expressed by a freely drawn curve in the plane. Dirichlet said y is called a function of x if y possesses one or more definite values for each of certain values of x in an interval X_{0} to X_{1}. But today a function is a subset of a Cartesian product in which the first element of an ordered pair occurs only once; that is, a function is a restricted set of ordered pairs of numbers; it is single valued. Added to this changing concept we have the introduction of the new word "relation," which is far more inclusive than function. The nature of a function—in particular, the linear, quadratic, exponential, and logarithmic functions—will be treated as much as operations with them. Function will be distinct from a relation, and both will be related to sets of ordered pairs of numbers, and a rule or set builder. The distributive law may be cited as an example of meaningful algebra. This law is the basic idea behind both mental and written arithmetic, the use of parentheses, factoring, multiplication of polynomials, and the manipulations of fractions. If the law is understood, most special methods of treating these topics can be eliminated. Algebra will be further enhanced by introducing deductive reasoning—a procedure that should be taught in all courses of school mathematics and not in geometry courses alone. Thus geometry can be relieved of the burden of total responsibility for deductive methods. For example, students will make certain assumptions and definitions, accept certain undefined terms, and then prove theorems, for example, the square of an odd number is an odd number, and that it is always one more than a multiple of eight. Through such deductive procedures diverse bits of information become related and "hang together," thereby promoting understanding and assisting the memory. Further, such understanding contributes to the ability to use algebra to solve problems. Throughout the presentation the concepts and language of set theory will be used. The set concept is elementary and closely related to experience. It permits a variety and richness of problems that call for creative and original thinking, and is one of the great unifying and generalizing concepts of all mathematics. Meaning becomes of utmost importance. Geometry. The program in geometry will be vastly different from the present program. The usual one and onehalf years devoted to plane and solid synthetic geometry will be reduced to considerably less than one year of study. Solid geometry as a halfyear course of deductive methods will disappear entirely. Those aspects of solid geometry of importance will be developed along with the corresponding concepts of plane geometry. The treatment of the sphere can be coordinated with the circle: locus will be treated simultaneously in two and three space. The mensuration and construction problems of solid geometry will be developed on an intuitive rather than the present pseudodeductive basis. Since deduction is to be stressed throughout the study of mathematics it is not necessary to spend a year on deducing theorems and originals of synthetic geometry. In fact the study of plane synthetic geometry will begin with an informal and intuitive introduction to geometric ideas, followed by a discussion of the nature of deductive reasoning. The formal study will then start with the postulation of the congruence theorems, and proceed as rapidly as possible through a chain of six or eight fundamental theorems to the proof of the Pythagorean theorem. Onethird of a year of study is sufficient for this. With the Pythagorean theorem established, it is possible to proceed to analytic geometry where the fundamental ideas of distance, division of a line segment, slope, equation of a line, and equation of a circle are developed. Thus, a new powerful geometry exists for the student to use. To prove or deduce theorems and originals, both analytic and synthetic methods are now at his disposal. Trigonometry. Vectors is another topic of great importance both in physics and in further mathematical study. A unit on vectors, with its assumptions and undefined terms, and the ideas of displacement, multiplication by a scalar, addition, and subtraction will give the student an added tool for proving theorems in geometry and solving problems in forces, acceleration, and rectilinear motion. The study of plane vectors also gives an approach to trigonometry that permits the easy development of periodicity properties of the trigonometric functions. Today, trigonometry has become an integral part of analysis and has hardly any support as a separate subject, isolated for a half year of study. The trigonometry of real numbers has been developed as a wrapping function of the real axis, and the sine function then repeats its values periodically every 2π, for example, f (2π + x) = f(x). Further the sine is an odd function that is sin (x) = sin (x), while the cosine is an even function that is cos (x) = cos (x). This behavior permits the application of trigonometry to all sorts of periodic phenomena, such as light and sound waves, alternating current, business cycles, heat flow, and harmonic analysis. The concepts of amplitude, period, frequency, and phase, in relation to the sine and cosine as functions of real numbers, have had significant effects in all branches of pure and applied mathematics. The study of trigonometry will from the very start be related to coordinates, both rectangular and polar. It is to be noted that coordinates are to be continually used throughout the entire study of mathematics in the curriculum of the future. This is a new emphasis. The solution of triangles will play a minor role, the solution of oblique triangles being limited to the use of the law of sines and the law of cosines, without the use of logarithms. The logarithmic solution of triangles, a speedy method fifty or more years ago, has been outmoded with the advent of electric calculators. The analytic aspects of trigonometry will be emphasized. In the past, in applications to surveying and navigation, the solution of plane and spherical triangles was the central theme. In the newer uses of trigonometry it is the circular functions of real numbers that occupy the central theme. The study of these functions is greatly facilitated by the concept of a function as a set of ordered pairs of numbers. The usual formulas will be developed with the use of coordinate geometry and in the spirit that trigonometry is a study of periodic functions of real numbers. Twelfth Year. The elimination of parts of traditional algebra, and much of Euclidean geometry, gives us time to complete the high school program with a study of those topics of advanced algebra that can properly be classified as modern analysis. There will be a study of the elementary functions—polynomial, rational, exponential, logarithmic, and inverse functions. This leads to study of absolute value and limits, followed by a unit on polynomial calculus developed from a meaningful and conceptual viewpoint and not one that is formalistic and mechanical. I would add one more item of study to the high school curriculum of the future. Just where and how it is to come into the curriculum must be worked out. This is the topic of probability and statistics. Many of the newer applications of mathematics are those underlying chance, that is, probability and statistical inference. These applications involve problems in safety, genetics, longevity, industrial planning, cost of living, occupational choice, testing, measuring, public opinion poles, theory of games, and so on. Statistical thinking is playing a greater and greater role in the life of educated men and women. An introduction to statistical inference with probabilistic conclusions is as important as the study of deductive systems with universal conclusions. The two important aspects of statistics are: (1) in a distribution of attributes arising from chance causes there is a coexistence of stability (a central tendency) accompanied by a variation; and (2) by proper sampling techniques based on mathematical probability theory, both the stability and the variation of the entire population can be predicted to a high degree of confidence. The study of the first aspect is accomplished in descriptive statistics, that is the collection, organization, and analysis of numerical data through the use of frequency tables, graphs of distribution, the arithmetic mean, mode, and median, the range, and standard deviation. The second aspect is acquired through the study of combinatorial analysis and partitioning of sets. Both these aspects will be in the mathematics curriculum of the future. SUMMARY OF THE CURRICULUM The aim of this entire curriculum is to enable those who are capable and desire to study science, mathematics, technology, or engineering, to enter college prepared to begin a rigorous freshman course in Differential and Integral Calculus. Realizing this aim will be a necessity for the development of our country's scientific progress in the immediate years ahead of us. All highly capable students should pursue this study. For the other collegebound students, the more advanced part of the proposed curriculum may be delayed and studied as a firstyear college program such as that proposed by the College Undergraduate Program Committee of the Mathematical Association of America. Any seventeenth century mathematician reappearing upon earth today could enter most classrooms in our high schools and, without any preparation, teach the present traditional curriculum, so far is it behind the times. But in order to handle the curriculum proposed here, the seventeenth century mathematician would be at a total loss and have to bring himself up to date with respect to fundamental concepts in every field included in the curriculum—algebra, geometry, trigonometry, calculus, statistics, logic. Most important of all, he would have to catch the spirit of modern mathematics, which began in the twentieth century, a spirit wherein we seek patterns of thought, mathematical forms, rather than specific tricks. It is in this sense that the proposed curriculum is modern and for the future. Finally, I should like to add only one word concerning those in high school who are not capable or not planning to go to college. Many of the present courses and textbooks for these students are a rehash and stew of everything under the sun. There is no organization, structure, or systematic development of mathematics in many of the books or proposed curricula. It is my hypothesis that the mathematics for these students will and must be the same as the elementary portions of the curriculum I have outlined. The mathematics used in general education in enabling us to understand our universe and solve our daily quantitative problems is the same mathematics that the scientist uses in his research into nature. The difference is one of complexity and depth. It is merely longer time and more concrete illustrations that are needed for the slow learner, and not a different type of curriculum. PREPARING TEACHERS FOR THE NEW PROGRAM Many teachers are convinced that arguments for curricula reform are valid, and that modernization of the curriculum is not only in order but long overdue. These teachers strongly desire to bring their programs and their teaching up to the highest possible standards. At the same time they ask, Who is sufficient to do these things? I shall try to give a brief answer. Only a very small per cent of the present teachers can possibly have had uptodate training in subject matter required for the task. Only those who have begun their teaching careers very recently have had an opportunity to take college courses modern in content, and even most of these teachers have had collegiate training of the traditional character. Many teachers are asking, What is contemporary mathematics? Where can we learn it? How can we use it in our classes? New materials must be made available for classroom teachers, new or greatly revised textbooks must be written, and manuals for teachers must be provided. It is difficult to make changes in educational procedure. It is easy to teach what we were taught in the way we were taught. It is difficult and troublesome to teach something new to which we have heretofore never been exposed. Up to now, mathematicians have gone on their way, creating new concepts and points of view, but not communicating these to the secondary school teacher. Year after year, high school teachers have taught the nineteenthcentury program in a satisfactory way, not studying or searching for newly discovered mathematics. There has thus resulted a huge gap between the frontiers of mathematical knowledge and the high school program. We must now close this gap. The immediate concern of teachers is the study and acquisition of mathematical knowledge in the following areas. 1. Modern analysis, including the contemporary concepts of variable, function, relation, coordinates and lattices, sentences, and inequalities. 2. Modern algebra, including the basic theory of sets, groups, rings, fields, matrices, linear algebra, and vector space. 3. Modern geometry as a set of transformations, including the basic structure and elements of projective, affine, Euclidean and nonEuclidean geometries, as well as finite systems. 4. Symbolic logic, including the contemporary theory of axiomatics and nature of mathematical proof. 5. Probability, from a settheoretic approach, including the study of trees, partitioning, combinatorial analysis and continuous, as well as discrete, data. 6. Statistics, both descriptive and inferential, including sampling theory, tests of hypotheses, tests of significance, and design of experiments. When the teacher has mastered the elements of these areas of contemporary mathematics, then the teaching will of necessity be modern and enthusiastic. As a fitting close to this discussion, I quote a statement made by a prominent Russian mathematician and educator, B. V. Gnedenko, in 1957. A teacher who reduces his task to the point that he communicates to the pupil only the sum of knowledge specified in the curriculum and merely teaches the pupil to deal with routine problems, rarely achieves any success. From the teacher is demanded enthusiasm for his subject and the conviction that his subject is one of the most important affairs of the nation. From the students a love for mathematics and a conviction of the creative powers of his students in mathematics; and that he describe in general outline before their intellectual gaze, the impressive picture of the uninterrupted development of mathematics, with its close relations to technology, the natural sciences, and all the other manifestations of human activity.3 Can we require any less of ourselves in the years that lie ahead? 1 This article is based on a talk given at the Conference on the American High School, held at the University of Chicago. 2 Dr. Fehr is president of the National Council of Teachers of Mathematics and a member of the Mathematical Association of America and the American Mathematical Society. He is the author of Secondary Mathematics and coauthor (with Veryl Schult) of Arithmetic at Work and Arithmetic in Life, all published by D. C. Heath. 3 American Mathematical Monthly, Vol. 64, No. 6, JuneJuly 1957.


