From Mark Alder:  “I am very grateful for the kind permission of Professor Kline's widow, Mrs. Helen M. Kline, to use this material on this section of my website.” 

Copyright © Helen M. Kline & Mark Alder 2000

Preface

For many generations the United States maintained a rather fixed mathematics curriculum at the elementary and high school levels. This curriculum, which we shall refer to as the traditional one, is still taught in fifty to sixty per cent of the American schools. During the past fifteen years a new curriculum for the elementary and high schools has been fashioned and has gained rather wide acceptance. It is called the modern mathematics or new mathematics curriculum. Though many groups have contributed to it and their recommendations are not quite identical, for present purposes I believe that it is proper and fair to overlook the differences among them.

The experimental work on the new curriculum, such as it was, has been done. Hundreds of new texts have been written and millions of children and young people have been and are being taught with this new material. In addition, several dozen books have been published which explain the new curriculum to parents, teachers, principals, superintendents, and other interested parties. The money, time, energy, and thought expended on this program have been considerable -even enormous.

Mathematics occupies a centra1 position in the schools. Students spend eight years on it in the elementary schools and from two to four years in the high schools. - Moreover, the subject has proved to be an obstacle to scholastic achievement for many students. Hence the question of whether the new curriculum has actually improved the teaching of mathematics and has indeed made the subject more accessible to the students is important.

Now that the new program has been somewhat stabilized and its nature made clear, it seems possible and necessary to decide whether progress has actually been made. Are our chíldren really better off by reason of this nationwide, highly touted reform? Certainly the education of our children is too important for us to accept a curriculum uncritically just because it has been extensively proinoted and has been backed by many professors of mathematics.

Up to the present time it has been assumed by the genera1 public that the profession has spoken and that the ouly problem is how to extend the teaching of the new curriculuin to more and more schools. However, sharp difierences of opinion as to the merits of the innovations do exist among professional mathematicians and teachers. It behooves all interested parties to examìne the effectiveness of the new material lest the innovations become established as the new orthodoxy despite the absence of any firm evidence that the innovations are genuine improvements. This is what I propose to do.

I hope that the reader wifl feel, as I do, that any book critical of a particular attempt at reform is not ipso facto reactionary. The traditional curriculum has major defects and I shall cite them. It needs to be improved. But it seeins to me that true progress is possible - and a truly progressive attitude can exist - only if we have the courage to admit that any particular attempt at reform has not worked.

I am indebted to many people for helpful criticisms and suggestions but especially to Professor Fred V. Pohle of Adelphi University, Professor Alexander Calandra of Washington University, and to Dr. George Grossman, Director of Mathematics for the New York City Board of Education. I am a1so indebted to Mr. Thomas McCormack, president of St. Martin's Press, not only for criticisms and suggestions, but for his encouragement to publish this book. He stressed repeatedly that a critique of the mathematics curriculurn would be a public service. Of course the particular views expressed herein are chargeable only to myself.

Morris Kline

1973

CHAPTER 1 - A Taste of Modern Mathematics.

".. . Great God! I'd rather be

A Pagan suckled in a creed outworn;

So might l, . .

Have glimpses that would make me less forlorn."

William Wordsworth

Let us look into a modern mathematics classroom. The teacher asks, "Why is 2 + 3 = 3 + 2?"

Unhesitatingly the students reply, "Because both equal 5"

No, reproves the teacher, the correct answer is because the commutative law of addition holds. Her next question is, Why is 9 + 2 = 11?

Again the students respond at once: "9 and 1 are 10 and 1 more is 11."

"Wrong," the teacher exclaims. "The correct answer is that by the definition of 2,

9 + 2 = 9 + (1 + 1).

But because the associative law of addition holds,

9+(1+1)=(9+1)+1.

Now 9 + 1 is 10 by the definition of 10 and 10 + 1 is 11 by the definition of 11."

Evidently the class is not doing too well and so the teacher tries a simpler question. "Is 7 a number?" Thà students, taken aback by the simplicity of the question, hardly deem it necessary to answer; but the sheer habit of obedìence causes them to reply affirmatively. The teacher is aghast. "If I asked you who you are, what would you say?"

The students are now wary of replying, but one more courageous youngster does do so: "I am Robert Sinith."

The teacher looks incredulous and says chidingly, "You mean that you are the name Robert Smith? Of course not. You are a person and your name is Robert Smith. Now let us get back to my original question: Is 7 a number? 0f course not!  It is the name of a number. 5 + 2, 6 + 1, and 8 - 1 are names for the same number. The symbol 7 is a numeral for the number.

The teacher sees that the students do not appreciate the distinction and so she tries another tack. "Is the number 3 half of the number 8?" she asks. Then she answers her own question: "Of course not! But the numeral 3 is half of the numeral 8, the right half."

The students are now bursting to ask, "What then is a number?" However, they are so discouraged by the wrong answers they have given that they no longer have the heart to voice the question. This is extremely fortunate for the teacher, because to explain what a number really is would be beyond her capacity and certainly beyond the capacity of the students to understand it. And so thereafter the students are careful to say that 7 is a numeral, not a number. Just what a number is they never find out.

The teacher is not fazed by the pupils poor answers. She asks, "How can we express properly the whole numbers between 6 and 9?"

"Why," one pupil answers, "just 7 and 8."

"No", the teacher replies. "It is the set of numbers which is the intersection of the set of whole numbers larger than 6 and the set of whole numbers less than 9."

Thus are students taught the use of sets and, presumably, precision.

A teacher thoroughly convinced of the vaunted value of precise language, and wishing to ask her students whether a number of lollipops equals a number of girls, phrases the question thus: "Find out if the set of lollipops is in one-to-one correspondence with the set of girls." Needless to say, she gets no answer from the students.

Bent but not broken, the teacher asks one more question: "How much is 2 divided by 4?"

A bright student says unhesitatingly, "Minus 2."

"How did you get that result?" asks the teacher.

"Well," says the student, "you have taught us that division is repeated subtraction. I subtracted 4 from 2 and got minus 2."

It wouId seem that the poor children would deserve some relaxation after school, but parents anxious to know what progress their children are making a1so query them. One parent asked his eight-year-old child, "How inuch is 5 + 3?" The answer he received was that 5 + 3 = 3 + 5 by the commutative law. Flabbergasted, he re-phrased the question: "But how many apples are 5 apples and 3 apples?"

The child didn't quite understand that "and" means plus and so he asked, "Do you mean 5 apples plus 3 apples?"

The parent hastened to say yes and waited expectantly.

"Oh," said the child, "it doesn't matter whether you are talking about apples, pears or books;

5 + 3 = 3 + 5 in every case."

Another father, concerned about how his young son was getting a1ong in arithmetic, asked him how he was faring.

"Not so well," the boy replied. "The teacher keeps talking about associative, commutative and distributive laws. I just add and get the right answer, but she doesnt like that.

These minor examples may illustrate, and perhaps caricature, some features of the currìculum now called modern mathematics or the new mathematics. We shall examine the major features in greater detail in due course and we sha1l consider their merits and demerits. But first, we shall revíew briefly the old mathematics to see what defects prompted the development of a new curriculum.

CHAPTER 2 - The Traditional Curiculum

"I have found you an argument but I am not obliged to find you an understanding."

Samuel Johnson

Though the traditional curriculum has been affected somewhat in recent years by the spirit of reform, its basic features are readily described. The first six grades of the elementary school are devoted to arithmetic. In the seventh and eighth grades the students take up a bit of algebra and simple facts of geometry such as formulas for area and volume of common figures. The first year of high school is concerned with elementary algebra, the second with deductive geometry, and the third with more a1gebra (generally called intermediate a1gebra) and with trigonometry. The fourth high schooi year usually covers solid geometry and advanced algebra; however, there has not been as much uniformity about fourth-year work as there has been for the earlier years.

Several serious criticisms of this curriculum have been voiced repeatedly. The first major criticism, which applies to algebra in particular, is that it presents inechanical processes and therefore forces the student to rely upon memorization rather than understanding.

The nature of such mechanical processes can readily be illustrated. Let us consider an arithmetical example. To add the fractions 5/4 and 2/3, that is, to ca1cu1ate

students are told to find first the least common denominator, that is, the smallest number into which 4 and 3divide evenly. This number is 12. One then divides 4 into 12, obtains 3, and mu1tiplies the numerator 5 of the first fraction by 3. Siniilarly one divides 3 into 12, obtains 4, and multiplies the numerator 2 of the second fraction by 4. The result thus far is to convert the above sum into the equal sum

One now sees easily that the sum is 23/12.

A good teacher would no doubt do his best to help students grasp the rationale of this process, but on the whole the traditiona1 curricu1um does not pay much attention to understanding. It relies upon drill to get, students to do the process readily.

After students learn to add numerical fractions they face a new hurdle when asked in a1gebra to add fractions where letters are involved. Though the saime process is used to ca1cu1ate

the individual steps are more complicated. Again the curriculum relies upon drill to put the lesson across. The students are asked to carry out the additions in nurmerous exercises until they can perform them readily

They are taught many dozens of such processes: factoring, solving equations in one and two unknowns, the uses of exponents, addition, subtraction, multiplication and division of polynomials, and operations with negatlve numbers and radicals such as \/3 in each case they are asked to imitate what the teacher and the text show them how to do. Hence the students are faced with a bewildering variety of processes which they repeat by rote in order to master them. The learning is almost always sheer memorization.

It is a1so true that the various processes are disconnected, at least as usually presented. They rarely have much to do with each other. While all these processes do contrìbute to the goal of enabling the student to perform a1gebraic operations in advaned mathematics, as far as the students can see the topics are unrelated. They are like pages torn from a hundred different books, no one of which conveys the life, meaning and spirit of mathematics. This presentation of algebra begins nowhere and ends nowhere.

After a year of such work in algebra the traditiona1 curriculum shifts to Euclidean geometry. Here mathematics sudden1y becomes deductive. That is, the text starts with definitions of the geometrìcai figures and with axioms or basic assertions which are presumably "obviously true" about the figures. They then prove theorems by applying deductive reasoning to the axioms. The theorems follow each other in a logical sequence; that is, the proofs of later theorems depend upon the conclusions already established in the earlier theorems. The sudden shift from mechanical algebra to deductive geometry certainly bothers most students. They have not thus far in their mathematics education learned what "proof" is and must master this concept in addition to learning subject matter proper.

The concept of proof is fundamnental in mathematics, and so in geometry the students have the opportunity to learn one of the great features of the subject. But since the final deductive proof of a theorem is usually the end result of a lot of guessing and experimenting and often depends on an ingenious scheme which permits proving the theorem in the proper logical sequence, the proof is not necessarily a natural one, that is, one which would suggest itself readily to the adolescnt mind. Moreover, the deductive argument gives no insight into the difficulties that were overcome in the original creation of the proof. Hence the student cannot see the rationale and he does the same thing in geometry that he does in algebra. He memorizes the proof.

Another problem troubles many students. Since algebra is also part of mathematics, why is deductive proof required in geometry but not in algebra? This problein becomes more pointed when students take intermediate algebra, usually after the geometry course, because there proof is again abandoned in favor of techniques.

With or without proof, the traditional method of teaching results in far too much of only one kind of learning - memorization. The claim that such a presentation teaches thinging is grossly exaggerated. By way of evidence, if evidence is needed, I have challenged hundreds of high school and college teachers to give open book examinations. This suggestion shocks them. But if we are really teaching thinking and not memorization, what could the students take from the books?

The traditiona1 curriculum has also become too traditional. Some topics that received considerable emphasis for generations have lost significance but are still retained. One example is the solution of triangies in trigonometry. Here, given some parts - sides and angles of a triangle, the theory shows how to.compute other parts and even how to use logarithms in the calcu1ations. This topic, which had far more relevance when trigonometry was taught primarily to prospective surveyors, should have been deemphasized long ago. Another example is the computation of irrational roots of polynomia1 equations. The method usually taught, called Horner's rnethod, requires several weeks of class tinre and does not warrant it.

There are also minor logical defects in the traditional currìcuium. For example, students are taught that x2-4 can be factored into (x+2) (x-2), but that x2-2 cannot be factored. However, the latter can be factored if we are willing to introduce irrational numbers. In this event the factors are x-\/2 and x +\/2 . Likewise x2 + 4 can be factored if we are willing to use complex nurnbers. In this case the factors are x + 2i and x-2i

where i =\/-1 . Thus the error made in the traditional method of teaching is the failure to specify the class of numbers we are willing to consider in order to perform the factorìng.

Beyond the few defects we have already described, the traditional curriculum suflers from the gravest defect that one can charge to any curriculum - lack of motivation. Mathematics proper, to use the words of the famous twentieth-century mathematician Hermann Weyl, has the inhuman quality of starlight, brilliant and sharp, but cold. It is also abstract. It dea1s with mental concepts, though some, such as geometrical ones, can be visualized. On both accounts, the coldness and the abstractness, very few students are attracted to the subject.

Young people can no doubt see that there is some point to learning arithnretic but they can see little reason to study algebra, geometry and trigonometry. Why shou1d they learn the addition of algebraic fractions, the solution of equations, factoring and other topics? The appeal of geometry is not greater. It is true that students can see what geometry is about and what the theorems assert; the figures make clear what this branch of mathernatics dea1s with. But the question of why one should study this material is still not answered. One can readily understand what the history of China is about, but may still question why he shou1d be obliged to learn it.Why is it important to know that the opposite angles of a parallelogram are equal or that the altitudes of a triangle meet in a point?

Clearly one cannot defend algebra, geometry and trigonometry on the ground that they will be of use later in life. The educated layman does not have occasion to use this knowledge at any tinre unless he becomes a professional scientist, mathematician, or engineer. But this group cannot be more than a few per cent of the high school popu1ation. Moreover, even if all of the students were to use some mathematics later in life this usage cannot be motivation. Young people cannot be asked to take seriously material that they might need years later. This motivation is often described as offering "pie in the sky."

As a matter of fact, in an effort to motivate the students, the schools did try to teach some uses of arithmetic in the seventh and eighth grades. They taught simple and compound interest and discount on loans. But twelve- and thirteen-year-old students did not take to such material and the experinrent is conceded to be a failure. The motivation must appeal to the student at the time he takes the course.

Another motivation often dang1ed before students is that they must study mathematics to get into college. If the mathematics they have been taught in elementary and high school is typical of what lies ahead in college, they may not want to go to college.

The prospective mathematicians, scientists, and engineers will find mathematics useful in their careers. But if the mathematics presented gives no inkling of how it will be useful and if it is in itself totally unattractive, telling the students that it is needed in science and engineering wiIl only encourage them to seek another career.

Much of the mathematics taught is often defended as "training the mind." There may very well be some training, but the same effect can be achieved with subject matter that is far more understandable and agreeable. One could teach the commonly used forms of reasoning by resorting to social or simple legal problems whose relevance to life is far more apparent to the students. One does not need mathematics to teach people that the statement "All good cars are expensive" is not the same statement as "All expensive cars are good." Moreover, the use of social or legal problems does not require the mastery of technical language, symbolism, and abstract concepts, which tend to obscure the reasoning. Thus it is far more difficult for the student to see that the statement "All parallelograms are quadrilaterals" is not the same as "All quadrilaterals are parallelograms." In fact, experience in teaching shows that to make the logical arguments used in mathematical reasoning clear to the student, one must use non mathematical examples involving the same arguments. Moreover, there is some question about whether the training to think in one sphere carries over to thinking in another. One may be inclined to believe that it does, but one could not prove that this is so.

Another commonly advanced justiification for teaching mathematics at the high sohool level is the beauty of the subject. But we know that the subjects taught have not been selected because they are beautiful. They have been selected because they are necessary for further work in mathematics. There is no beauty in adding fractions, in the quadratic formula, or in the law of sines. No amount of preaching or rhapsodizíng about the beauty of mathematics will make such ug1y ducklings appealing. Moreover, novitiates - are not likely to find beauty in a subject they are still striving to master, any more than one who is learning French granrmar can appreciate the beauty in French literature.

A few students are attracted to mathematics by the intellectual challenge or because they like what they happen to do well. The rare student who experiences this challenge may indeed be intrigued - as some mathematicians are - by the fact that there are only five regular polyhedra. However, as far as most students are concerned, the world would be just as well off if there were an inifinite number of them. As a matter of fact, there is an infinite number of regular polygons and no one seems depressed by this fact.

There is indeed an intellectual value in mathematics. But there is a question of whether young people can appreciate it just as there is about whether a six-year-old can appreciate Beethovens music. If the teacher proves a theorem of mathematics, the student will still be struggling to understand the theorean, its proof and its meaning. While undergoing such struggles the student is not likely to be impressed with the intellectual content and what the human mind has accomplished. In him the theorem and proof produce bewilderment and confusion.

Beyond the purported values of training the mind, beauty and intellectual challenge, the defenders of the traditional curriculum point to the exercises. These, they say, show uses of mathematics and should convince the student that the material is important. There are work problems such as the ditch-digger's dilemma. "One man can dig - a ditch in two days and another in three days. How much time will be required if both men dig it together" Such problems create pointless work.

Then there are tank-filling problems for students who have no swimming pools to fill. Or the mixture problems: "How many quarts of milk with ten per cent cream and how many quarts of milk with five per cent cream must be mixed to make a hundred quarts of milk with fifty per cent cream?" Such problems are useful to farmers who wish to fake the cream content of their milk. Other mixture problems concern mixing brands of coffee or brands of tea to make undrinkable brews.

There are age problems too: "Jane is twenty years older than Mary. In ten years time Jane will be twice as old. How old is Mary?" This type of problem calls for finding out other people's ages, and many people are sensitive about their ages.

There are also number problems, such as "One number is three times another number minus two. What are the numbers?"(The numbers racket is actually illegal.) More realistic are board problems. "A board seven feet long is to be cut into two parts, one of which is to be two feet longer tban the other. How long are the parts?" Of course students are bored with board problems.

And we shouldnt neglect to mention the time, rate and distance problems, such as up- and down-river travel for students who are going nowhere and whose desire to go anywhere has not been aroused. Some problems involve taking walks around a circular garden and ask for the dimensions of the garden. If we allowed the students to take walks around the garden and provided each with a pleasant companion we wou1d do the stu-dents more good.

All these problems are hopelessly artificial and will not convince anyone that algebra is usefu1.

Some authors of algebra texts do point to "truly physical" problems. For example, Ohms law states that the voltage E equals the current I times the resistance R. In symbols E = IR. Calculate E if I = 20 and R = 30.

But the current involved in such problems doesnt drive any mental motors. So far as the student is informed, Ohms law could be describing the number of marriages in Burma each year.

For generations the calculus textbooks have asked students to calculate centers of gravity and moments of inertia of bodies without ever pointing out why these quantities are significant. Consequently, the gravity of these problems produces nothing but inertia in the students. Such physical problems, presented with no preliminary explanation of physical background or physica1 significance, mean nothing to the student. Clearly, a physical application is worthless if the student cannot see what is accomplished.

Even the use of the word "application" is often bothersome. Students are taught, say, a formula for area and are then asked to calculate areas with it. These calculations are supposed to be an application. This kind of application adds insult to injury. Since the so-called applications are still pointless and still part of mathematics proper, in what sense are they applications?

The fact is, then, that no motivation for the study of mathematics is offered in the traditiona1 curriculum. Students take it because they are required to. Motivation means more than a psychological stimulus. Genuine motivatîon also supplies insight into the very meaning of the mathematics. A great deal of mathematics, particularly on the elementary level, was suggested directly by real situations and problems. The bare formula s = 16t2 acquires meaning when one learns that it relates the distance fallen and time of travel of an object which is dropped. An ellipse becomes more than just another curve when one learns that it is the path of a planet around the sun. Moreover, the questions that are raised about the formula and about the curve become meaningful because they concern the physical situations. The physical meanings also supply, in many cases at least, the power to think about the mathematical problems that are raised, because the mathematics is no more than a representation of the physics and a means of solving physical and other problems.

The failure to present the meaning of mathematics is analogous to teaching students how to read musical notation without allowing them to play the music. Students might be taught how to recognize full notes, half notes, sharps, flats, the key, and how to transpose music from one key to another without ever hearing any music. But if they do not hear what these various notations and techniques mean, they will be left with meaningless and boring skills.

The traditional curricu1um has been faithfully reproduced in thousands of textbooks. The strongest reaction induced by the traditionaltexts is that they are insufferably. dull. Most textbook writers seem to believe that scientific writing must be cold, spiritless, mechanical and dry. These books have no authors. They are not only printed by machines; they are written by macbines.

Textbook writers also seem to take inordinate pride in brevity, which can often be interpreted as incomprehensibility. Reasons for steps are either not given or given so briefly as to make the presentation almost useless to the student. Many authors seem to be saying, "I have learned this materia1 and now I defy you to learn it."Brevity in mathematical exposition is the soul of witlessness and obscurity.

The most disturbing fact about many traditional mathematics texts is that they lack originality and repeat each other endlessly. A few thousand arithmetic, algebra, geometry, and trigonometry texts have been published since 1900. Practically all of the texts on any one of these subjects contain the same materia1 and presentation; only the order of the topics is different.

But there is hope for progress because each contains at least ten topics, and the number of permutations of ten objects ten at a time is 3,628,800. It would be difficu1t to estimate how many trigonometry texts have been written with the justification that they treat the general angle before the acute angle. One can be sure, however, that just as many boast of treating the acute angle before the general angle. The only thing that is acute about these books is the pain they give the reader.

Are there no variations among these books? There are variations such as the elementary algebra and the advanced algebra, the elementary advanced algebra and the advanced elementary algebra, the half-course and the full course, the seven-eighths course, and so forth. Here, too, there is hope for "progress" because there are irrational numbers; hence, we can look forward to irrational algebra courses.

What is especia1ly disturbing about these books is that many of the authors are consciously dishonest to their profession. I asked one professor who had written "umpteen" trigonometries of the full and partially full type why he included such useless topics as the solution of oblique triangles by the law of tangents and the law of half-ang1es. He admitted that these topics are worthless, but said he included them because the books sell better. Apparent1y, no matter how many trigonometries a man may write, not even one can reflect his honest judgment.

I asked another professor, who published a stereotyped college a1gebra, why he bothered to write such repetitious nonsense. Oh, he said, I can write the stuff between classes without having to think about it. Why shouldn’t I do it? Needless to say, no thinking was evident in the presentation of the material.

Another professor published a book which included some material that he believed to be unimportant. He admits this in his preface and then says quite candidly that he included this material with an eye to the market. Such honest dishonesty!

Those authors who repeat each others topics are in a sense plagiarizing. But the plagiarism extends beyond that. Paraphrases of whole sections of material covering many paragraphs are readily found. One author took whole chapters from another book with on1y minor changes, acknowledging, of course, the inspiration of God, Euclid, Newton, and Einstein.

Most traditiona1 mathematics textbooks appear to be commercial jobs that make a contribution only to the authors' pocketbooks. The ethics of some teachers, to say nothing of their mentalities, is evidently in a sad state. The only persons who can claim any credit for original work in connection with these books are the publishers' publicity men, who must think up good blurbs for the advertisements.

Fairness requires that one mention recent improvements in the format of mathematics texts. Important formulas are now enclosed in red boxes. Other texts use overlays of plastic to show the increasing complexity of a figure as one overlay after another is superimposed on the origina1 text figure.

Clearly the defects of the traditiona1 curriculum are numerous. The reliance upon memorization of processes and proofs, the disparate treatments of algebra and geometry, minor logical defects, the retention of a few outinoded topics, and the absence of any motivation or appeal explain why young people do not like the subject and therefore do not do well in it. Their dislike is intensified and their difficulties in understanding are compounded by being asked to read dull, poorly written, and commercially contrived textbooks.

Certainly reform was called for. The leaders of the new mathematics movement did not cite all of the above defects. However, they did point the finger at some of them. So let us look now at what these people proposed to do and try to evaluate their effectiveness in improving the teaching of mathematics.

CHAPTER 3: The Origin of the Modern Mathematics Movement.

"Experience, however, shows that for the majority of the cultured, even of scientists, mathematics remains the science of the incomprehensible".

Alfred Pringsheim

There was general agreement in the early 1950s and even before that date that the teaching of mathematics had been unsuccessful. Student grades in mathematics were far lower than in other subjects. Student dislike and even dread of rnathematics were widespread. Educated adults retained almost nothing of the rnatheinatics they were taught and could not perform sirnple operations with fractions. In fact, these people did not hesitate to say that they got nothing out of their mathematics courses. When this country entered World War 11, the military discovered quickly that the men were deficient in mathematics and that they had to institute special courses to bring up the level of proficiency.

Thõugh there are many factors that determine the outcome of any teaching activity, the groups that undertook reform focused on curriculum and argued that if this component were improved the teaching of mathematics would be successful.

In 1952 the University of Illinois Committee on School Matheinatics headed by Professor Max Beberman began fashioning a new, or modem, mathematics curriculum.

By 1960 the curriculum (at that time directed solely toward the high schools) was used on an experimental basis. Subsequently, the Committee undertook to provide an elementary school curriculum and gradually extended the teaching of both the elementary and high school subject matter to additional geographical areas. The experimental texts, in photo-offset form, were eventually published as commercial texts.

In 1955 The College Entrance Examination Board, whose function is to prepare college entrance examinations which meet the requirements of many colleges, decided to take up the problem of the high school mathe-matics curriculum and compose what it considered to be the proper one. It set up its own Cominission on Mathematics. In 1959 the Commission îssued its report, Program for College Preparatory Mathematics, and adjoined severa1 appendices which contained samples of recommended subject matter. It did not produce texts. During the years 1955 to 1959 and for severa1 years thereafter, the members of the Commission toured the country and campaigned for the kind of curriculum it proposed in its Program.

In the fall of 1957 the Russians launched their first Sputnik. This event convinced our government and country that we must be behind the Russians in matheinatics and science and had the effect of loosening the purse strings of governmental agencies and foundations. It may be coincidence but at this tiine many oher groups decided to go into the business of producing a new curriculum.

The American Mathematical Society, the organization concerned with research, decided in 1958 that its talents shou1d be applied to the fashioning of a high school curriculum, and it set up a new group, The School Matheniatics Study Group, headed by Professor Edward G. Beg1e, then at Yale University, to undertake the task. This group began its work by writing curricula for the junior and senior high schools and then extended its program to include the elementary school arithmetic curriculum.

The Nationa1 Council of Teachers of Mathematics set up its own curriculum committee, The Secondary School Curriculum Conimittee, which came out with its recommendations in an article in the May 1959 issue of The Mathematics Teacher. Many other groups, such as the Ball State Project, the University of Maryland Mathematics Project, the Minnesota School Science and Mathçmatics Center, and the Greater Cleveland Mathematics Program, were soon formed and began their work.

Individual high school and college teachers commenced in the late 1950s to write their own texts along the lines already foreshadowed or explicitly recommended by the curriculum groups. By the early 1960s a spate of such books had appeared, and many more have continued to appear since that time.

Rather surprisingly, the many groups and independent textbook writers all headed in about the same direction. Hence they have all, fairly enough, been described by the term "modem mathematics" (or "new mathematics").

The origin of the term modern mathematics is relevant. Even before the members of the Commission on Mathematics had determined just what they were going to recommend, they gave addresses to large groups of teachers. Their main message was that mathematics education had failed because the traditional curriculum offered antiquated mathematics, by which they meant mathematics created before 1700. Implicit in this con-tention was the assumption that young people were aware õf this fact and therefore refused to learn the material. Wou1d you, argued these educators, go to a lawyer or a physician whose knowledge of his profession was limited to what was known before 1700? Though these speakers were presumably informed in mathematics they ignored completely the fact that mathematics is a cumulative development and that it is practicaliy impossible to learn the newer creations if one does not know the older ones. Nevertheless, the Commission contended that we must drop the traditional subject matter in favor of such newer fields as abstract algebra, topology, symbolic logic, set theory, and Boolean algebra. The slogan of reform became "modern mathematics".

As it turned out, the reform offered as much a new approach to the traditiona1 curriculum as it did new contents, and some groups emphasized this fact. Hence the term modern mathematics is not really an appropriate description of the new curricula. However, perhaps because the propaganda value of the word modern was too useful to drop - a 1970 automobìle is c1early more desirable than a 1969 model - the terms modem mathematics or new mathematics have been retained.

While the modern or new mathematics curriculum as it stands today was being fashioned by the groups already mentioned, new groups appeared on the scene and began to recommend more radica1 reform. For example, an intemationaI group meeting at Royaumont, France, in 1959 urged the abandonment of virtually all the familiar courses in high school mathmatics, including Euclidean geometry. The conference declared these subjects to be outdated by electronics, relativity, computers and the soaring importance of abstract mathematics as the basìs of modern science. The new subjects were to be logic, structure, and the unity of mathematics as a whole and were to be taught in a new language This conference did not resu1t in the formation of another curriculum group, but it encouraged still furtber departures from the traditional curriculum.

Of the newer groups which have proposed more radical reforms we shall mention two. In the summer of 1963 a group of mathematicians assembled for The Cambridge Conference on School Mathematics. (Its report, Goals for School Mathematics, appeared as a publication by the Houghton Mifflin Company.) This group recommended the inclusion -by the end of grade twelve, the fourth year of high school - of many additional advanced topics drawn from the theory of numbers, abstract algebra, linear algebra, n-dimensional geometry, projective geometry, tensors, topology, differential equations, and of course, the calculus. On page 7 the report asserts,

"The subject matter which we are proposing can be roughly described by saying that a student who has worked through tbe fuIl thirteen years of mathematics in grades K to 12 (kindergarten through the fourth year of high school) should have a level of training comparable to 3 years of a top-level college training today".

The justification for advocating such a program when the already existing curriculum groups had barely begun to try out their programs or were still fashioning theiri was given in the Foreword by Francis Keppel, who was then United States Comnaíssioner of Education. He observed that recent curriculum changes are essentially different from those attempted in the past aad that the reforms have been eminently successful for the most part (How Dr. Keppel knew this in 1963 when most new mathematics curricula had hardly been tried is not clear.), so much so that

"it has sometimes been difficult to distinguish their shortcomings. Yet the shortcomings are there, and they are by no means insignificant. It cau be argued, in fact, that the deficiencies of the present reform movement are grave enough to threaten the expressed goals of the movements themselves".

Keppel then noted that the changes recommended by the Cambridge group were intended to represent the subject as the scholar saw the discipline, and that the students were assumed to be able to learn far more than they had been expected to in the past. The limitations of the teacher were noted too!

"Most curriculum reforms, practically enough, have chosen to limit their ambitions in the light of these realities. They have tended to create such new courses as existìng teachers, after enjoying the benefits of brief retraining, can competently handle. They have done so fully aware that they are thus setting an upper limit, and an upper limit that is uncomfortably close"

Keppel then continued:

"If the matter were to end there, the result might well be disastrous. New curricula would be frozen into the educational system that would come to possess, in time, all the deficiencies of curricula that are now being swept away. And in all likelihood, the present enthusiasm for curriculum reform will have long since been spent; the 'new' curricula might remain in the system until, like the old, they become not only inadequate but in fact intolerable. Given the relative conservatism of the educational system, aad the tendency of the scholar to retreat to his own direct concerns, the lag may well be at least as long as it has been during the first half of this century.

The present report is a bold step toward meeting this problem. It is characterized by a comlete impatience with the present capacities of the educational systein. It is not oniy that most teachers will be completely incapable of teaching much of the mathematics set forth in the curricula proposed here; most teachers would be hard put to comprehend it. No brief period of retraining will suffice. Even the first grade curriculum embodies notions with which the average teacher is totally unfamiliar.

None the less [sic], these are the curricula toward which the schools should be aimìng. . . ."

The second of the newer groups joining the movement to revi~e curricula, the Secondary School Mathematics Curriculum Improvement Study, was organized in. 1965 by Professor Howard Fehr of Columbia University. Its goal is to reconstruct secondary school mathematics from a globa1 point of view. It seeks to eliminate the barriers separating the severa1 branches of mathematics and to unify the subject through its general concepts, sets, operations, mappings, relations and structure. (We shall discuss these concepts later.) Professor Fehr's contention is that his organization of the subject matter will permit the introduction into the high school curriculum of much that has been considered collegiate mathematics. The work of the Cambridge group and of the Curriculum Improvement Study has proceeded slowly and their effect on the schools is not widespread as yet. Hence our account and evaluation of the modern mathematics movement wiIl concentrate on the curriculum efforts of the preceding groups, some of which are still at work on one aspect or another of the school programs.

The curricula which have been formulated by these several organizations are the product of group efforts in which research mathematicians, college and high school teachers and even representatives of industry have collaborated. On the face of it, such collaboration would seem to be a wise procedure. However, attempts to achieve a meeting of minds often result in compromises that are not satisfactory to anyone or which vitiate the thrust of. the effort. The point may be illustrated by the story that the famous dancer Isadora Duncan offered herself in marriage to Bernard Shaw and perhaps somewhat facetiously said, '. . . and think of the child who would have your brains and my looks. "Yes", said Shaw, but what if the child shou1d have your brains and my looks'

When one seeks to determine what changes these curricula offer, why these changes are desirable, and what reasoning or evidence can be proffered to support the desirabilîty of these changes, one is faced with a problem of considerable magnitude. It is true that in its 1959 report the Commission on Mathematics of the College Entrance Examination Board did describe the contents it recomniended. However, except for stressing that modern society requires a totally new mathematics the Commission did not defend the contents it proposed. Moreover, the various curriculum groups that did write texts not only extended the reform to the elementary school grades but did not necessarily follow the Cominissions recomendations. One wou1d have expected that each group wou1d have declared its own position and have presented its case for including or excluding particular topics and for adopting its own approach. No such documents have been issued. This is aIl the more true of the many texts published by individual authors which proclaim themselves to be modem in character. Hence we are left to infer for ourselves what the modem mathematics curriculum is and why it is presumably superior to the traditional curricultun. Could the absence of explanatory and justifying material be interpreted to mean that the advocates of modem mathematics are not too clear themselves on where they have headed, or are they fearful that explicit statements of the features and purported merits of their materials will not bear scrutiny? In any event, to determine the nature and qualities of the modern mathematics curriculum, one must examine the texts and listen to the speeches made by various proponents. At the moment, pending a fuller discussion, let us note that there are two main features of the new curriculum: a new approach to the traditiona1 mathematics, and new contents.

Since we intend to evaluate the new mathematics, it is necessary to consider on what basis one should judge it. One cou1d use as a criterion, Is the mathematics correct? The answer is yes, but the criterion is useless. Correctness does not guarantee that the students will take to the material, that they can absorb it, or that this particular mathematics is what should be taught.

Will it develop mathematicians? Even if it were the idea1 curriculum for the training of mathematicians one could not be content. The new mathematics is taught to elementary and high school students who will ultimately enter into the full variety of professions, businesses, technical jobs, and trades, or become primarily wives and mothers. Of the elementary school children, not one in a thousand will be a mathematician; and of the academic high school students, not one in a hundred will be a mathematician. Clearly then, a curriculum that might be ideal for the training of mathematicians would still not be right for these levels of education.

The contents should contribute to the goals of elementary and high school education and should be accessible to young people. The approach to the material should make the content inviting and aid comprehen. sion as far as possible. In particular the new mathematics should remedy at least some of the defects of the traditional curriculum. Unfortunately, in the field of education, unlike mathematics proper, one cannot give an ironclad proof that a particular principle or topic is right or wrong. But there are arguments which do enable us to decide.

Though a dozen or more groups have fashioned new curricula and by now many series of new mathatics texts are on the market, we have already noted that they all adopted about the same approach and contain about the same material. This uniformity has resulted in part from imitation. It is also a consequence of the emphasis and direction which mathematicians are favoring in current research and which we shall discuss at greater length later. Hence, though not every statement we shall make about the new mathematics applies necessarily to any one curriculum, it is fair to treat them as a single movement characterized by common features and content.

We intend to consider carefully the nature of the new mathematics prograxn and to discuss its merits and demerits. Before doing so we should like to inject a somewhat different but nevertheless relevant criticism. Reform of mathematics education was called for, but there is a serious question as to whether curriculum was the weakest component and should have been tackled first. It would, I believe, be generally conceded that the policy of universal education pursued in the United States is highly commendable, but our country was not and still is not prepared to carry on such a .program. Certainly we do not have enough qualified teachers; therefore the education in many parts of this country is woefully weak. Were more good teachers available they wou1d have been able long ago, by acting in concert, to remedy the defects of the traditional curriculum. Since the teacher is at least as important as the curriculum, the money, time and energy devoted to curriculum reform might well have been devoted to the improvement of teachers. It is true that in 1958 the National Science Foundation inaugurated and has maintaìned various institutes for the education of teachers. These institutes should have been used to improve the mathematìcal backgrounds of elementary and high school teachers so that they cou1d form more independent judgments of what is important in mathematics. Unfortunately, they have been used largely to teach teachers how to teach mathematics of unproven worth.

Whether or not curriculum reform shou1d have received priority, the historical fact must be faced that the new curriculum is at hand and is being widely used. Let us therefore attempt to evaluate it.

Chapter 4: The Deductive Approach to Mathematics.

The great science [mathematics] occupies itself at least as much with the power of imagination as with the power of logical conclusion.

Johann Friedrich Herbart

One of the major criticisms of the traditional curriculum is that students learn to do mathematics by rote, by memorizing procedures and proofs. It is the contention of the advocates of the modern mathematics curriculum that when the subject is taught logically, when the reasoning behind steps is revealed, students will no longer have to rely upon rote learning. They will understand the mathematics. The logical approach is, in other words, also the pedagogical approach and the panacea for the difficu1ties students have had in learning mathematics.

Just what does the logical approach mean? Basica1ly it is the one commonly used in the tradidina1 curriculum to teach high school geometry. That is, one starts with definitions and axioms and proves conclusions, called theorems, deductively. Though this approach has been used in geometry, it has not been used in the teaching of arithmetic, algebra, and trigonometry. Hence, so far as this feature of the new curriculum is concerned, the major change is in these latter subjects. Let us see what the deductive approach to arithmetic and algebra entails.

[* The reader who is familiar with the deductive approach may want to skip the examples given below]

The examples we shall examine are taken from typical modern mathematics texts.

The approach to arithmetic usually presupposes that we know what the numbers 0, 1 ,2, . . ., called the counting numbers, are. It then introduces axioms. We know that 4 added to 3 yields the same number as 3 added to 4, or 3 + 4 = 4 + 3. This axiom is stated in symbolic language as

a + b = b + a and is called the commutative axiom. That is, we may commute or interchange the order in which we add two numbers.

If we had to ca1cu1ate 3 + 4 + 5 we cou1d ca1culate 3 + 4 and then add 5 to the result or we could calcu1ate 4 + 5 and add this result to 3. That is, we cou1d first associate the 3 and 4 and then add 5 or we cou1d first associate the 4 and 5 and add the resu1t, 9, to the 3. Thus we have the associative axiom of addition. In symbolic language it states that

(a + b) + c = a + (b + c). The commutative and associative axioms apply to multiplication aIso. That is, a x b = b x a and (a x b) x c = a x (b x c).

Let us consider next 3 x (4 + 5). This is 3 x 9 or 27. It is also 3 x 4 + 3 x 5. That is, a x (b + c) = a x b + a x c. This fact is another axiom and it is called the distributive axiom. That is, we may distribute the

mu1tiplication over the b and the c instead of applying it to b + c.

There are other axioms. For example, the sum and product of two counting numbers is a unique counting

~ The reader who is familiar with the deductive approach may want to skip the examples in the next few pages.

number. There is a unique number 0, such that a + 0 =a for each counting number a, and there is a unique

number 1 such that a x 1 = a for each counting number a.

These axioms are used to justify steps in arithmetic. Consider the addition of 38 and 3. This is performed as follows.

By the definition of 38

38 + 3 = (30 + 8) + 3.

The associative axiom tells us that

(30+8) + 3 = 30 + (8+3).

Now, by the definition of 3,

30 + (8+3) = 30 + (8 + [2 + 1])

and by the associative axiom for addition

30 + (8 + [2+1] ) = 30 + ([8 + 2] + 1).

Addition of 8 and 2 yields

30 + ([8 + 2] + 1) = 30 + (10 + 1).

By the associative axiom

30 + (10 + 1) = (30 + 10) + 1,

and - since 30 + 10 = 40 and 40 + 1 = 41, we have

finally deduced that

38+3=41.

To see how the distributive axiom is used in arithmetic let us consider

7 x 1 3. By the definition of 13

7 x 13=7 x (10 + 3).

Now, the distributive axiom tells us that Since 7 x 10 = 70 and

7 x 3 = 21, we have deduced that -

7 x 13 = 91.

Presumably these steps, which employ the distributive axiom, make clearer how the 91 is arrived at than by mu1tiplying in the usual fashion wherein one says that 7 x 3 = 2 1, writes down the 1, carries the 2 over to the tens, column and adds it to the result obtained from multiplying 7 and 1.

It is true that our method of writing numbers such as 13 employs what is called positional notation. That is, the 1 in 13 stands for 10 and this must be taken into account in any method of teaching mu1tiplication. Whether citing the distributive axiom clarifies the operation is the issue. Let us suppose that it does and see what it leads to. Consider the problem of 17 x 13. To follow the. above pattern we must do as follows:

17 x 13 = (10 + 7) x (10 + 3).

Now 10 + 7 must be regarded as a single number and by the distributive axiom

(10 + 7) x (10 + 3) = [10 + 7] x 10 + [10 + 7] x 3.

By the commutative axiom of multiplication

[10 + 7] x 10 + [10 + 7] x 3=l0 x [10 +7 ] + 3 x [10 + 7],

and again by the distributive axiom

l0 x [10+7]+3 x [10 + 7]

= (10 x 10 + 10 x 7) + (3 x 10 + 3 x 7).

Ca1cu1ation of the quantities in the parentheses and addition of the two results yield 221. We can readily envision the steps that would have to be undertaken to multiply, say, 172 by 135.

At some stage in the development of arithmetic and algebra, usually between the seventh and the ninth grades, students are asked to learn negative numbers.To motivate the introduction of negative numbers students are asked first, what number x satisfies the equation 17 + x = 21 Here the answer is clearly 4. Now comes the crucial question. What number x satisfies the equation 21 + x = 17?

To answer this question, one writes 21 as 17 + 4 so that

(17 + 4) + x =17.

By the associative axiom

17 + (4 + x) = 17.

But by the definition of 0

17 + 0 = 17,

so that, since the zero is unique,

4+x=0.

We see that if there were a number x such that x + 4 = 0 then we would be able to solve the original equation. We are therefore motivated to introduce the number -4 with the understanding that 4 + (-4) = 0.

The counting numbers (except 0) are now called the positive integers and the new numbers are called the negative integers. To operate with negative integers the axioms or basic properties that apply to the counting nusnbers are now assumed to hold for the combination of the old counting numbers and the new negative numbers. Thus, since the cominutative, and associative axioms of addition hold for the counting numbers, they hold for the positive and negative integers.For example, to add -2 and -5 we note first that by the definition of -2 and -5,

(-2 + 2) +(-5 + 5) = 0 + 0 = 0.

But by the commutative and associative axioms

(-2 + 2) + (-5 + 5) = 2 + 5 + [(-2) + (-5)1, -

so that

0 - 7 + [(-2) + (-5)].

Hence -2 + (-5) must be -7 because when it is added to 7 it gives 0.

Having Iearned how to add negative numbers and posi-tive and negative numbers students are told that sub-, tracting a number means adding its inverse. 4 is the inverse of —4 and conversely. Hence

17 - 13 = 17 + (-13) = 4

6 - 8 = 6 + (-8) = -2

-5 - (-11) = -5 + 11 = 6.

Before establishing additiona1 properties of negative nuxnbers, let us prove that a x 0 = 0 for every integer a. Since by the definition of 0,

a + 0 = a, then

a x (a + 0) = a x a.

However, by the distributive axiom

a x (a+0) = a x a + a x 0.

Hence

a x a + a x 0 = a x a.

Therefore

a x 0 = 0,

because 0 is that number which when added to any number gives b.Now students are presumably prepared to appreciate the proof that a negative integer times a positive integer is negative. That is, -3 x 4 = -12. We start with

(1)............................a x [b + (-b)]=a x 0 = 0

because b + (-b) = 0. However, by the distributive axiom

(2)...............................a x [b + (-b)] = a x b + a x (-b).

Hence the right sides of (1) and (2) are equa1, so that

a x b + a x (-b) = 0.

Then a x (-b) must be -(a x b) because a x (-b) added to a x b gives 0 and this is true only for the additive inverse of a x b, namely, -(a x b).

Finally, if in place of (1) we start with

-a x [b + (-b)]

and carry through the same steps with —a replacing a, we can prove that

-a x -b = a X b.

To introduce signed (positive and negative) numbers another method is often used. One starts with the num-bers, 1,2,3,4, . . .; these are the counting numbers except for 0, and are called the natural numbers. Then an integer is defined as the equivalence class of ordered pairs of natural numbers. What this means is the following. An ordered pair of natural numbers is the pair (7,5). This, intuitively, means 7 - 5. However, (6,4), (4,2), (8,6), and millions of other pairs represent the same integer. Two such ordered pairs (a,b) and (c,d) are called equiva1ent if a + d = b + c. Hence the integer 2 is the class of all ordered pairs eqilivalent to, say (7,5). The merit of this definition is that one can, using on1y the natura1 numbers, introduce the ordered pair (5,7), which intuitively represents 5 - 7, or -2. Again (5,7), (4,6), (6,8), and so on are the same nega-tive integer, -2. The integer which we usua1ly denote by 0 is the class of ordered pairs (5,5), (6,6), (7,7), and so on. In the previous method we had to create the negative numbers. In the present method we construct the negative numbers.

Under this approach the operations with positive and negative integers are defined in terms of the ordered pairs. Thus, the sum of (7,5) and (6,3) is (13,8). Intuitively we have 2 + 3 = 5, but the logica1 development calls for the former. More generally

(a,b) + (c,d) = (a + c, b + d). We see that for the negative integers the definition of sum works in the same way. Thus (5,7) + (3,6) = (8,13) or, intuitively, -2 + -3 = -5. The ordered pair definition can be used to introduce stubtraction, multiplication and division of integers, and miraculously one obtains the usual laws for handling positive and negative integers.

To introduce fractions an approach similar to that used to introduce negative numbers is employed. The student is asked to find the value of x for which 3x = 6. Clearly x = (1/3) x 6 or 6/3. He is then asked to find the x for which 3x = 7. No integer will satisfy this equation. Hence we create new numbers, the fractions. In particular we create for x the number 7/3, which means (1 /3) x 7, and we agree that 3 x 1/3 = 1. Having introduced the fractions we agree further that the commutative, associative and distributive axioms are to apply to them. We can then prove that the usual operations for the addition, subtraction, multiplication and division of fractions apply.

In the case of fractions, too, some texts introduce them as ordered pairs of signed integers. Thus (3,5) is a fraction which intuitively means 3/5. The operations are defined in terms of the ordered pairs and then one can prove that addition and multiplication are commu-tative, associative and distiibutive.

The deductive approach encounters a serious obstacle, at least on the elementary level, in treating. irrational numbers. The reader may recall that numbers such as and the like are also members of the number system. The full logical treatment of. these nfimbers is much too difficult to be understood by tyros, and even if the students could assimilate the material the time re-quired to teach it would be inordinate. Hence the texts usually comproxnise. They "motivate" the introduction of such numbers by first showing that to solve

x2=4

we take the square root of both sides and obtain x = ±2.

Similarly to solve

x2 = 2

one takes the square root of both sides and obtains x = ± \/2.

Thus one is led to introduce the irrational numbers. One can indeed prove, and many texts do, that 2 is not a rationa1 number; that is, it is not equai to a quotient of two integers. So it is clear that objects such as 2 are new kinds of numbers. But the "motivation" used above does not suffice to introduce all irrationa1 numbers,  in particular. It also does not serve as a logical basis on which to build the properties of irrational numbers. Consequently these properties have to be postulated. For

example, it is true, if a and b are greater than zero, that  But since it cannot be proved on an elementary level the texts just assert it and give it a name. It is called the product property of square roots. Likewise the fact that cannot be proved, and so is merely asserted and labeled the quotient property for square roots.

Other difficulties arise in the introduction of complex numbers, that is, numbers of the form  etc. The new curriculuni, in this case like the traditional one, usually invents as the solution of

x2 = -1 and then forms the complex numbers. Alternatively it introduces complex numbers as ordered pairs of real numbers. It then defines the arithmetic operations with complex numbers and proves that the associative, commutative and distributive properties apply to these operations.

The logical foundation of arithmetic now serves to build algebra. Algebra is distinguished from arithnietic in that it deals with expressions involving letters such as 3x2 + 5x + 2 and operates with such expressions. Since the letters stand for numbers and all numbers obey the same laws, the letters obey these laws. Let us consider one algebraic proof. We shall prove that if a x b = 0, then either a is 0 or b is 0 or both are zero.

If a = 0 the theorem is proved. If a does not equal zero then a property of the number system tells us that there is an inverse, 1/a. Then, since

a x b = 0 and any number mu1tiplied by 0 yields 0,

Some of the textbook authors wish to cut short the logica1 development of the number system or to "innovate". The authors of one widely used text, in introducing negative numbers, list many of the usual properties:

the sum of any two integers is an integer, the commutative, associative, and distributive properties and such peculiar ones as the property of the opposite of a sum:

-(a + b) = -a + (-b).

They give examples of the uses of these properties which students are asked to imitate. Some thirty or forty properties are finally listed and the students are expected to learn and apply them. The authors do not say whether these properties are axioms nor do they prove the assertions. The net effect is confusion about what is proved from axioms and what are rules. In effect the authors are handing down rules despite their claims to be teaching deductive mathematics.

Our examples of the way in which deduction is applied to arithmetic and algebra have been very simple. One can readily imagine how lengthy and involved the proofs are when doing more complicated arithmetic and algebra.

The deductive approach to Euclidean geometry is essentially the one most adults learned in high school. Hence it is not necessary to illustrate it here. However, we shall have more to say about proof in geometry in the next chapter.

Since the major innovation of the new mathematics is the deductive approach to traditional subject matter, let us try to determine what pedagogical merit it may have. In particular, does it impart understanding of mathematics?

A number of considerations oblige us to answer this question negatively. First, let us examine how mathematics itself developed and see whether this history furnishes any evidence favoring one conclusion or another. After all, mathematics was created by human beings who certainly understood the subject. How did the masters Euclid, Archimedes, Newton, Euler, and Gauss come to understand mathematics?

Mathematics in a significant sense begins with the contributions of the Egyptians and Babylonians during the period of roughly 3000 to 300 B.C. These two peoples created the rudiments of arithinetic, algebra, and geometry. In arithmetic they worked with the positive whole numbers and fractions. Negative numbers were unknown to them and even the zero was not introduced despite the fact that Babylonians used positional notation in base sixty to write Iarge numbers. That is, in their syinbolism a number such as 125 meant 1.(60)2 + 2.60 + 5, just as in our base ten 125 means 1.102 + 2.10 + 5. Such a system of writing quantity ahiiost cries out for a zero, because to write 105 one needs the 0 to indicate that the 1 is not in the "tens" position but in the "hundreds". Yet the Babylonians did not create the zero. Their numbers were ambiguous and one had to judge from the context as to what was intended.

In geometry all that the Egyptians and Babylonians could manage were formulas for perimeter, area and volume of simple geometrical figures. For any figure pre-senting difficulties, such as the perimeter and the area of a circle, the formulas were only approximately right. Thus for almost three thousand years, two civilizations rather highly developed in fields such as art, religion, commerce, astronomy, and architecture got no further in mathematics than the rudiments. Moreover, they accepted all their resu1ts on a purely empirical basis. The concept of deductive proof was never even entertained. Could these civilizations have profited from more extensive knowledge of mathematics? Undoubtedly. All we can conclude is that more sophisticated mathematics does not come easily to human minds. In fact, eveu the little the Egyptians and Babylonians achieved is unusual compared - to what hundreds of other civilizations that had as much opportunity and need created in mathematics.

The first civilization in which mathematics can be said to have flourished is that of the classical Greeks; this civilization reached its zenith from 600 to 300 B.C. There is no question that the Greeks were of an unusual, even amazing, cast of mind. The classical Greek thinkers were indifferent to the needs of commerce, navigation, and practical matters generally, but they were intensely concerned about understanding the workings of nature. For this purpose they found geometry most suitable and it is in this area that they made their supreme contribution. The Greeks are also the people who first conceived of deductive mathematics. The goal was to obtain truths about nature and their plan was to start with some self-evident truths such as that two points determine a line and that all right angles are equal. Given these self-evident truths, or axioms, they planned to establish conclusions or theorems deductively. The theorems, then, would also be truths.

They did succeed in erecting several masterful structures, the foremost of which is Euclid's Elements, and this is the substance of the traditional high school geometry.course. However, Euclidean geometry did not come into being in this deductive manner. It took three hundred years, the period from Thales to Euclid, of exploration, fumbling, vague and even incorrect arguments before the Elements could be organized. Thus the Elements is the finished and relatively sophisticated product of much cruder, intuitive thinking. Even this structure, intended to be strictly logical, rests heavily on intuitive arguments, pointless and even meaningless definitions and inadequate proofs, as the nineteenth century mathematicians realized. What is most relevant, however, is that this deductive system came about after the understanding of all that went into it was achieved. Moreover, it is no accident that Euclidean geometry was the first subject to receive any extensive logical development; the reason is that the intuition can be readily applied to infer geometrical facts and the very figures suggest methods of proof.

It is also relevant that irrational numbers such as and the like were not accepted as numbers during the highest period of Greek culture. Why not? Because the whole numbers and fractions had an obvious physical meaning whereas the irrational numbers did not. The only intuitive meaning that one could attach to irrationa1s was that they represented certain geometrical lengths, such as the diagonal of a square whose sides are 1. What then did the Greeks do? They rejected

irrationals as numbers and thought of them as lengths. In fact they converted all of algebra into geometry in order to work with lengths, areas and volumes that might otherwise have to be represented numerically by irrational numbers and they even solved quadratic equations geometrically.

The history of mathematics subsequent to the high period of Greek culture is quite the opposite of what one ordinarily conceives mathematics to be. The progress that was made in the use of irrational numbers is due to Alexandrian Greek civilization, which was a fusion of the classicai Greek, Egyptian and Babylonian civilizations, and to the Hindus and Arabs who were entirely empirically oriented. It was the Hindus who decided that and their argument was that these irrationals could be "reckoned with like integers", that is, like Since the latter is obviously correct, as can be seen by taking the square root of each number, then so is .

Irrationa1 numbers were gradually accepted because of their utility and because familiarity breeds uncriticalness.

Negative numbers, introduced by the practical-minded about A.D. 600, did not gain acceptance for a thousand years. The reason: they lacked intuitive support. Some of the greatest mathematicians, Cardan, Vieta,Descartes, and Fermat, refused to work with negative numbers. The history of complex numbers is somewhat sinilar though they did not appear until about A.D. 1540 and only about two hundred years were required for these numbers to be used somewhat freely. A remark of the supreme mathematician Carl Friedrich Gauss is very pertinent. As is well known, he was one of the men who discovered the geometrica1 representation of complex nunmbers, and about this he said in 1831,

"Here [in this representation] the demonstration of an intuitive meaning of \/-1 is completely grounded and more is not needed in order to admit these quantities into the domain of the objects of arithmetic". Neither Descartes, Fermat, Newton, Leibniz, Euler, Lagrange, Gauss, or Cauchy could have given a definition of negative numbers or complex numbers, or irrationals for that matter. Yet all of them managed to work with these numbers quite satisfactorily, at least so far as their times employed these numbers. The history of the entire number system is pertinent not only because it shows how it was developed but also because algebra and analysis (the ca1cu1us and higher branches built on it) obviously utilize the number system, and whatever basis there was for the number system had to serve as the basis for a1gebra and analysis.

In the realm of algebra history has another significant tale to tell. The use of a letter to represent a fixed but unknown number dates from Greek times. However, the use of a letter or letters to stand for a whole class of numbers was not conceived of until the late sixteenth century. At that time François Vieta introduced expressions such as ax + b where a and b can be any (real) number..* The great merit of such genera1 expressions is that whatever we can prove about them is correct for aIl values of a, b, and x. Thus if we learn to solve the quadratic equation ax2 + bx + c = 0, we can solve all quadratic equations, because a, b, and c can stand for any numbers. The use of letters to stand for any numbers or even a restricted class of numbers is certainly, then, an enormous contribution and a seemingly simple one once it is pointed out. Yet during all the centuries that the Babylonians, Egyptians, Alexandrian Greeks, Hindus, and Arabs worked in algebra the idea of using letters for a class of numbers did not occur. These peoples did their a1gebra by working with concrete expressions such as 3x2 + 5x + 6 = 0. That is, they a1ways used numerical coefflciènts, and in fact most did not even use a symbol such as x for the unknown. They used words.

Why was the use of letters for genera1 coefficients so long delayed? The answer would seem to be that this device constitutes a higher level of abstraction in mathematics, a level farther removed from intuition. It is more difficult to think about ax2 + bx + c = 0 than about

3x2 + 5x + 6 = 0. Yet to reason deductively about the algebraic procedures for significant general expressions became possible only after these genera1 coefficients were introduced.

The history of the calculus is equally instructive. We shall not enter into the details of the concepts that lie at the foundation of that subject. But it may be sufficient to point out a few facts about its development. The great names in the creation of the calculus are, of course, Isaac Newton and Gottfried Wilhehn Leibniz. However, preceding them, Descartes, Fermat, Cavalieri, Pascai, Roberval, Barrow, and at least a dozen other well-known men made significant contributions. Despite the fact that so much was done before they did their work, neither Newton nor Leibniz could formulate correctly the basic concepts of the calculus. Newton wrote three major papers on the calculus and he put forth three editions of his masterpiece, The Mathematical Principles of Natural Philosophy. In each of these he gave a different explanation of the basic concept now called the derivative. Not one of these would be accepted from a beginning students of the calculus today. Leibniz in many papers was equally - unsuccessful. His first paper was described by the famous

Bernoulli brothers, James and John, as "an enigma rather than an explication".

There were many attacks on the work of both Newton and Leibniz. Newton did not respond but Leibniz did. He objected to "overprecise critics" and he argued that we should not be led by excessive scrupulousness to reject the fruits of invention. However, the defects were there and the attacks continued throughout the eighteenth century. The difflculties in clarifying the basic concepts of the calculus were so great that the famous eighteenth-century mathematician Jean LeRond d'Alembert had to advise students, "Persist and faith will come to you".

In view of the vague, unclear and even incorrect foundations of the calculus one might expect that the subject would collapse. But before the adequate deductive structure was created not only had the ca1culus been extended and successfully applied but the vast subjects of ordinary and partial differential equations, the calculus of variations, differential geometry, and the theory of functions of a complex variable had been erected on the calcu1us. How did mathematicians achieve these tremendous creations? Clearly they thought intuitively. Physical argu-ments, pictures, generalizations based on simple known cases, and experience with mathematics all helped them to draw correct conclusions.

It is highly significant that the logical foundations of the number system, algebra, and analysis (the calculus and its extensions) were not erected until the last part of the nineteenth century. In other words, during the centuries in which the major branches of mathematics were built up there was no Iogical development for most of it. Apparently the intuitions of great men are more powerful than their logic.

What can we infer from this history? It seems clear that the concepts which have the most intuitive meaning, the whole numbers, fractions, and geometrical concepts, were accepted and utilized first. The less intuitive ones, irrational numbers, negative numbers, complex numbers, the use of letters for general coefficients, and concepts of the calculus, required many centuries either to be created or to be accepted. Moreover, when they were accepted, it was not the logic that induced mathematicians to adopt them but arguments by analogy, the physical meaning of some concepts, and the obtainment of correct scientific results. In other words, it was intuitive evidence that induced mathematicians to accept them. The logic always came long after the creations and evidently was harder to come by. Thus the history of mathematics suggests, though it does not prove, that the logical approach is more difficult.

There are many who would base another argument against the deductive approach on the evidence of history. The argument is essentially that each person must go through about the same experiences that his

forefathers did if he is to attain the level of thought that many generations have achieved. This argument has been advanced by many great mathematicians who have concerned themselves with pedagogy. Henri Poincaré, one of the greatest mathematicians of recent times, said in his Foundations of Science (p. 437),

"The zoologists maintain that in a brief period the development of the embryo of an animal recapitulates the history of its ancestors of aIl geological epochs. It appears that it is the same in the development of the mind. The task of the educator is to make the mind of a child go through what his fathers have experienced, to pass rapidly through certain stages but not to omit any. For this purpose, the history of the science ought to be our guide".

The same thought has been expressed by one of the great mathematicians and teachers of the late nineteenth and early twentieth centuries. Felix Klein, in his Elementary Mathematics from an Advanced Standpoint (Dover reprint, 1945, Vol. 1, p. 268), says,

" In concluding thls discussion of the theory of assemblages [setsl we must again put the question which accompanies all of our lectures: How much of this can one use in the schools? From the standpoint of mathematical pedagogy, we must of course protest against putting such abstract and difficult things before the pupils too early. In order to give precise expression to my own view on this point, I should like to bring forward the biogenetic fundamental law, according to which the individual in his development goes through, in an abridged series, all the stages in the development of the species. Such thoughts have become today part and parcel of the general culture of everybody. Now, I think that instruction in mathematics, as well as in everything else, shou1d follow this law, at least in generai. Taking into account the native ability of youth, instruction should guide it slowly to higher ideas, and finally to abstract formulations, and in doing this it should follow the same road along which the human race has striven from its naive originai state to higher forms of knowledge. It is necessary to formulate this principle frequently, for there are always people who, after the fashion of the medieval scholastics, begin their instruction with the most general ideas, defending this method as the only scientific one. And yet this justification is based on anything but truth. To instruct scientifically can only mean to induce the person to think scientifically, but by no means to confront hlm, from the beginning, with cold scientiflcally polished systematics".

"An essential obstacle to the spreading of such a natural and truly scientific method of instruction is the lack of historical knowledge wbich so often makes itself felt. In order to combat this, I have made a point of introducing historical remarks into my presentation. By doing this I trust I have: made it clear to you how slowly all mathematical ideas have come into being, how they have nearly always appeared first in rather precursory form, and only after long development have crystallized into the deflnitive form so familiar in systematic presentation.

There is not much doubt that the difficulties the great mathematicians encounter are precisely the stumbling blocks that students experience and that no attempt to smother these difficulties with logical verbiage can succeed. If it took mathematicians a thousand years from the time first-class mathematics appeared to arrive at the concept of negative numbers - and it did - and if it took another thousand years for mathematicians to accept negative numbers - as it did - we may be sure that students will have difficulties with negative numbers. Moreover, the students will have to master these difficulties in about the same way that the mathematicians did, by gradually accustoming themselves to the new concepts, by working with them and by taking advantage of all the intuitive support that the teacher can muster.

One could of course argue that the growth of mathematics may indeed have proceeded as just described, but now that we have the proper logical structures for the number system, the calculus and other branches, we need not ask the students to repeat the fumblings of the masters. We can give students the correct approaches and they will understand them. This argument can be countered by the fact that the greatest mathematicians did try to build the logical foundations for the various subjects buf failed for centuries to do so. Their failure should serve as some evidence that the logical approaches are not easy to grasp. One can compress history and avoid many of the wasted efforts and pitfalls, but one caunot eliminate it. Of course, our students may be superior to the best mathematicians of the past.

But we shall not insist on the evidence of history. There are other weighty arguments against a purely deductive approach to elementary mathematics.

We might note, first of all, that we are not without experience in presenting mathematics deductively. Euclidean geometry has been presented in this manner for several centuries. Moreover, the intuitive meaning of this geometry is also evident to the student. Yet students have not been more successful in mastering geometry than algebra; nor have they left the geometry course with a feeling of elation because they have finally understood a branch of mathematics. If, then, the evidence of his-tory is not convincing, there must be other pedagogical arguments against a logical approach. They are not hard to find.

By the middle of the nineteenth century the various types of numbers and their properties were established on the basis of the uses made of them. Likewise, the properties of functions, derivatives and integrals used in the calcu1us were accepted on the basis of what seemed evident for the simplest functions or on the basis of the physical truth of the results obtained. The mathematicians then set about constructing logical foundations for the properties they had employed. In fact, the logic had to justify those properties, rather than determine them. Hence a very artificial and complicated structure of axioms and theorems was erected. The purpose of this structure was to satisfy the needs of professiona1 mathematicians who insist on deductive structure, but it was never intended as a pedagogical approach. Yet it is these logical foundations that the new mathematics employs to cultivate understanding.

The fact that utility determines the logical approach rather than the other way around is so basic in mathematics that the point warrants emphasis. Let us consider an example. To use addition of fractions in most rea1 situations, say, to add 1/2 ,and 1/3, we change both to sixths and then add 3/6 and 2/6 to obtain 5/6. However, when we multiply fractions we mu1tiply the numerators and multiply the denominators so that

1/2 x 1/3 = 1/6. We could "add" fractions by adding the numerators and adding the denominators and obtain 1/2 + 1/3 = 2/5. Why don't we use this latter inethod? It is simpler. But it does not fit experience. Having adopted a useful definition of addition the logical properties of addition must follow from the definition.

As another example we could consider matrix multiplication. It so happens that the uses to which matrices are put requires that the multiplication be noncommutative, though we could define a multiplication which is commutative. Since the multiplication must be noncommutative, the logical foundations of the theory of matrices must be suited to this fact. Therefore, logic does not dictate the contents of mathematics; the uses determine the logical structure. The logical organization is an afterthought and in a real sense is gilt on the lily.

[See Chapter 8, for the definit