The Mathematical Manpower Shortage

Mr. Belden, nationally known consultant on employee pension plans, also is Manager of the Southern California-Arizona Branch of The Union Central Life Insurance Company. This article is from an address at the 1956 Annual Meeting of the Conference of Actuaries in Public Practice, of which he is a member.

“Learning how to add is clearly an important step toward understand­ing of higher mathematics. And perhaps it also is the first step in mastering the fundamentals of economics.”

The shortage of people whose education and training make them suitable candidates to enter the scientific field is so well known that any citation of statis­tics in this regard would be repeti­tious and redundant.

We hear a great amount of talk about the necessity for channeling the gifted college student into sci­ence and mathematics. Such efforts have merit and should actively be pursued, but complete solution to the problem is not to be found at the college and university level. The truth is that many pupils reach the college level with a type of mathematical preparation which leaves them sorely unprepared to follow collegiate courses in mathe­matics or science. If an effective attack is to be made on the problem of critical shortages in scientific and mathematical personnel, it is going to be necessary to change completely the grand strategy of the attack. The firepower must start at the foundations of educa­tion.

In short, I believe we should start to train the child rather than the man. And we must make sure that the child is well trained.

If this process is followed, we will uncover the natural talent for scientific and mathematical sub­jects, which in many individuals is never awakened or realized, or in others is finally awakened, but at a time of life when to go back and fill the mind with elementals and fundamentals is distasteful, impractical, or perhaps even impossible.

Pythagoras, who lived in the sixth century B.C., was one of the first of the Greeks to develop an understanding of mathematics. He felt that his knowledge of arith­metic, geometry, and philosophy was so valuable that he would only teach it after swearing his pupils to secrecy and obtaining their agreement not to divulge the knowledge to others. Out of this came the Pythagorean brotherhoods which were disbanded in about 532 B.C. because of their re­ligious and political activity. How­ever, the importance of Pytha­goras’ mathematical teachings was recognized. In the following cen­tury there was no secrecy about the mathematics and philosophy which was being taught to Grecian youth by such men as Socrates and Plato.¹

According to Aristotle, who lived about 150 years after Pythagoras, the Pythagoreans “applied themselves to the study of mathematics and were the first to advance in that science; insomuch that, hav­ing been brought up in it, they thought that its principles must be the principles of all existing things.”²

Euclid’s Elements, first written on papyrus scrolls in about 300 B.C. and translated into Arabic per­haps 1,000 years later, became the basis of textbooks on geometry in modern languages. These were in wide general use until the early years of the present century.3

Progressive Education

There was a time prior to the rise of Progressive Education, under the aegis of John Dewey4 and his followers, when mathe­matics was required in some important and significant form at every level of the educational proc­ess, starting at the elementary grades and ending at the top as one of the rigid requirements for a Ph.D. Today the teachers’ colleges of many of our respected universities are granting the Ph.D. and the newer doctoral degrees without any requirements in mathematics.

The status of present-day mathe­matics at the secondary school level is illustrated by the case of the mother of a high school student who wrote to the editor of her newspaper to complain that when her child wanted to study algebra and geometry, the teacher and principal of the school had advised against it on the grounds that such courses would be of little value in later life. What retrogression since the time when Pythagoras taught that mathematics was the basis of all things!

Perhaps the change is, in part at least, explained by the following quotation from an opinion poll used at a “workshop” for teachers, administrators, and school board members at Teachers College, Co­lumbia University:

“Geometry and other branches of mathematics are not valuable for training persons to think….

“The newer types of school activi­ties—excursions, art, plays—should be emphasized even at the expense of a reduction in the time given to the three R’s.”

If the old arithmetic teaching methods were so wrong, why is it that the average high school grad­uate of past years knew his mul­tiplication tables without stopping to figure out the answers; could add and subtract with reasonable accuracy and speed; could do prob­lems in long division; could work out simple percentages and get the decimal in the right place; and had a basic knowledge of algebra and plane geometry? Today any employer of clerical help knows better than to expect a similar per­formance from the average re­cently graduated high school stu­dent. Two or three years after graduation they can do some of these things with reasonable pro­ficiency, but for the most part they have learned how to do the work on the job and after leaving school.

Why Wait for Deficiencies?

In considering the above situa­tion I am reminded of something I once read about the effectiveness of the “quickie” courses in reme­dial reading. If the backward reader can be brought up to standard by a few weeks of in­tensive training, why wait until he is backward to give him the inten­sive training? In arithmetic, why force our young people to spend the time from age 6 to 18 in school, only to have them find they must then intensively study what they were not taught in those 12 long years?

In June 1956 the Educational Testing Service of Princeton, N. J., published a booklet, Problems in Mathematical Education.5 This report opens with a quotation from the New York Times of June 20, 1955:

At the grassroots of our society, in the schools where our young peo­ple are being trained for their re­sponsibilities tomorrow, there are dry rot and decay which threaten not only the bright economic prospects before us but even our ability to re­main strong and defend ourselves. We need scientists, mathematicians, and engineers as never before, yet many teen-agers with the ability to assume such roles are simply not being given a chance to get the es­sential training. The statistics make plain that great mistakes have been made in thousands of communities throughout our land.

The report continues: “This is a serious indictment. It charges that the schools are falling down on an important part of their job, that at the very time we are increasingly dependent on competence in mathematics, many capable young­sters are denied the training they need. Recent studies tend to fortify this conclusion. Complaints from businessmen emphasize that even the most elementary skill, facility in ordinary arithmetic, is in short supply.”

The report of the Educational Testing Service then takes up the question of “The Learner.” In this section we find: “According to a recent national survey of high school seniors, 12 per cent of them had never taken any algebra or geometry; 26 per cent had quit studying mathematics after only one year, and another 30 per cent had dropped the subject by the end of the second year… 6 per cent of the brighter seniors (the top 30 per cent on a test of mental ability) do not give either algebra or geometry a try and another 41 per cent never get beyond the ele­mentary phases.”

The section of the report headed “The Teacher” is so important that I quote from it at some length: “One study showed that of 211 prospective elementary school teachers, nearly 150 had a long­standing hatred of arithmetic.

“This state of affairs may not be unrelated to a lack of ordinary competence with numbers. A ran­dom sample of 370 candidates for elementary school positions faced this question on an examination:

“The height of a letter in a cer­tain size of print is 1/₄ inch. If the following are the heights (in inches) of this letter in other sizes of print, which one is the next larger size?

“(a) 5/16 (b) 1/2 (c) 3/16 (d) 3/8 (e) 7/16

“Half of the 370 candidates picked a wrong answer.

“There have been numerous studies of teacher competence in arithmetic. If they can be believed, it seems pretty clear that many ele­mentary school teachers have a hard time keeping even half a jump ahead of their pupils.

“‘Elementary teachers, for the most part,’ according to one ob­server who has taught them, ‘are ignorant of the mathematical basis of arithmetic; high school teachers assigned to teach mathematics fall in this category also.’

“This ignorance is scarcely sur­prising, for little knowledge of mathematics is expected, even offi­cially, of prospective school teach­ers. In the majority of cases an in­dividual with ambition to teach in an elementary school can matricu­late at a teachers’ college without showing any high school mathe­matics on his record. He can gradu­ate without studying any college mathematics. And in this condi­tion, he can meet the requirements of most states for a certificate to teach arithmetic. The certification requirements for high school math teachers are only a little stiffer: nearly one-third of the states will license them even though they have had no college mathematics at all, and the average requirement for all states is only 10 semester hours.

“In the absence of data, let us accept the assumption that a teach­er who has a solid understanding of mathematics himself is more likely than not to develop a similar understanding in his pupils. Where does such an assumption take us? In a vicious circle, apparently—and in reverse—for it has been shown that ‘solid understanding’ is frequently absent in teachers. Fu­ture teachers pass through the ele­mentary schools learning to detest mathematics. They drop it in high school as early as possible. They avoid it in teachers’ colleges be­cause it is not required. They re­turn to the elementary school to teach a new generation to detest it.”

Start in Lower Grades

A great majority of the recent efforts to combat the mathematical manpower shortage have been di­rected at the collegiate level. While such activity is good, the problem will be solved only if attention is directed to mathematics training at every level of education, starting at the very lowest grades. The col­lege or university student who de­cides to pursue a career in science or mathematics will find he is unable to follow through with the idea if he has not received and ab­sorbed adequate primary and sec­ondary school mathematics. Also, there is low ratio of probability that a college freshman will have any worth-while or serious scien­tific or mathematical desire if his interest has not been aroused at the lower academic level.

Parents should demand that their children receive adequate drill in addition, subtraction, mul­tiplication, and division and that old-fashioned teaching techniques be returned to usage in the instruc­tion of these important elementary subjects.

Greece was the cultural and eco­nomic leader of the world in the days when the great teachers I have mentioned were at their height. These men were very con­scious of the value of mathematics; some thought it was the basis of all things. It was not until the Greek scholars began to lay their empha­sis on less important studies that Greece was overthrown and lost its power. The present Atomic Age and its nuclear physicists have proved that the Pythagoreans were not so far off the beam in their be­lief in regard to mathematics that “its principles must be the prin­ciples of all existing things.”

 

1. Encyclopaedia Britannica, 14th ed. Vol. 15, pp. 84-5; Vol. 18, pp. 48-50, 802-4. Hogben, Lancelot. Mathematics for the Million. Rev. ed. Norton, 1947. pp. 26-9, 30-1, 66-7, 163, 194-6; Wonderful World of Mathematics. Garden City Books, 1955. pp. 30-1. 33.

2. Encyclopaedia Britannica, 14th ed. Vol. 18, p. 803.

3. Ibid. Vol. 8, pp. 802-3. Hogben, Lancelot. The Wonderful World of Mathematics. Garden City Books, 1955. p. 33.

4. Dewey, John. The School and Society. Rev. ed. University of Chicago Press, 1915; Democracy and Education. Mac­millan, 1921; Experience and Educa­tion. Macmillan, 1938.

5. This report, based upon a study made possible by a grant from the Carnegie Corporation of New York, may be ob­tained from the Educational Testing Service, Princeton, N. J., for $1.00.

 

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The Responsibility of Education

In that terrifying novel by George Orwell, 1984, the Party of Big Brother developed the ultimate in ruthless dictatorship pre­cisely because it devised the means of enslaving men’s minds…. The crowning triumph of its torture chambers was the under­mining of the disciplines of logic and mathematics, by which it finally brought its victims not only to assert but actually to believe that two plus two equals five.

As yet, fortunately, it is only through fantasy that we can see what the destruction of the scholarly and scientific dis­ciplines would mean to mankind. From history, we can learn what their existence has meant. The sheer power of disciplined thought is revealed in practically all the great intellectual and technological advances which the human race has made. The ability of the man of disciplined mind to direct this power effec­tively upon problems for which he was not specifically trained is proved by examples without number. This ability to solve new problems by using the accumulated intellectual power of the race is mankind’s most precious possession. To transmit this power of disciplined thinking is the primary and inescapable responsibility of education.

Arthur E. Bestor, Jr., Education for 1984