Is Progress an Illusion?
By Lynn Arthur Steen
(Adapted from a talk given at the annual Silver and Gold banquet that concluded MathFest in Boulder, Colorado, on August 2, 2003.)
Lynn Arthur Steen
In his introduction [MAA president] Ron Graham joked that in contrast to the youthful energy of Project NExT, some wags have begun calling the Silver and Gold banquet Project Last. What I?d like to do for the next twenty minutes or so is to go one step further and present Project Past ? a glimpse at challenges facing our predecessors one hundred years ago. I do not intend to answer the question posed by my title, but instead to offer selected evidence that will help you decide for yourselves whether our sense of progress is real or illusory.
To focus this exercise, I looked only at the scientific literature of 1903 ? exactly one hundred years ago. Here are examples, arranged as they appeared from January through November; all but two are from Science,, the journal of the American Association for the Advancement of Science (AAAS):
Asaph Hall. ?The Science of Astronomy.? Science, 17:418 (January 2, 1903) 1-8.
Eliakim Hastings (E.H.) Moore. ?On the Foundations of Mathematics.? Science, 17:248 (March 13, 1903) 401-406.
G.A. Miller. ?Some Fundamental Discoveries in Mathematics.? Science, 17:430 (March, 27, 1903) 496-499.
Thomas M. Drown. ?From High School to College.? Science, 17:431 (April 3, 1903) 521-529.
Karl Pearson. ?Homogeneity and Heterogeneity in Collections of Crania.? Biometrika, 2:3 (June, 1903) 345-346.
S.W. Williston. ?Specialization in Education.? Science, 18:448 (July 31, 1903) 129-138.
Robert Simpson (R.S.) Woodward. ?Education and the World?s Work of Today.? Science, 18:449 (August 7, 1903) 161-169.
C.M. Woodward. ?The New Opportunity for Secondary Schools.? Science, 18:451 (August 21, 1903) 225-233.
National Education Association. ?Resolutions.? Science, 18:452 (August 28, 1903) 283-284.
Karl Pearson. ?On the Laws of Inheritance in Man.? Biometrika, 2:4 (November 1903), 357-462.
Three of the authors (E.H. Moore, G.A. Miller, and R.S.Woodward) were mathematicians; two (Moore and Woodward) were presidents of the American Mathematical Society. The others were scientists, well known in their time: Asaph Hall was an astronomer, Thomas Drown a chemist, Karl Pearson a statistician, S.W. Williston a paleontologist, and C.M. Woodward an engineer who served in 1902-03 as president of the American Society for Engineering Education.
Three of these turn-of-the-century papers review major fields of study: Hall on astronomy, Miller on mathematics, and Pearson on what we now call genetics. I think we can stipulate that subsequent progress in these fields is not an illusion. With one exception, however, the other papers in this sample deal largely with educational matters where the issue of progress becomes more problematic. Let?s imagine eavesdropping on a meeting in 1903 where leading scientists discuss educational problems. (Please bear with their masculine language which, although jarring to modern ears, was the language of the day.)
On educational practice: ?The consensus of public opinion regards education as a series of routine performances ... involving tasks which students sometimes undertake with joy and sometimes with sorrow and ending for those who complete the program with a ceremony called graduation. I think it would be troublesome to explain just what is accomplished by this process and why a person subjected to it may be called educated and one not so fortunate may be called uneducated.? (R.S. Woodward, p. 162)
On mathematics curriculum: ?In the schools algebra is taught in one water-tight compartment, geometry in another, and physics in another. [The] student learns to appreciate (if ever) only very late the absolutely close connection between these different subjects and then, if he credits the fraternity of teachers with knowing the closeness of this relation, he blames them most heartily for their unaccountably stupid way of teaching him.? (E.H. Moore, p. 410)
On school dropouts: ?Not one half of the boys and girls in Boston ever get inside a high school. ... It is not because the people are poor ? that excuse would cover but a small percent of the absentees ? [nor] because they have not ample brains and average common sense. In my judgment, the best word to explain the non-appearance of over fifty per cent of boys and girls in our secondary schools is ?incompatibility.? There is a lack of harmony. The school does not give what the pupils want. ?The average secondary school, if it prepares pupils for anything, prepares them for college; and since college is not for the majority, then secondary school is not for the majority. What then is there for the majority? If they are to have secondary education at all, it must be something different.? (C.M. Woodward, pp. 227-228)
If these complaints sound slightly familiar, so do some of the proposed solutions. Here are three resolutions from the National Education Association (NEA), all from 1903, published (perhaps surprisingly) in AAAS?s flagship journal Science:
On teacher pay: ?Teaching in the public schools will not be a suitably attractive and permanent career, nor will it command as much of the ability of the country as it should, until the teachers are properly compensated and are assured of an undisturbed tenure during efficiency and good behavior. ... The compensation of the teacher should be sufficient to maintain an appropriate standard of living. Legislative measures to give support to these principles deserve the approval of the press and the people.?
On taxation: ?The true source of the strength of any system of public education lies in the regard of the people whom it immediately serves, and in their willingness to make sacrifices for it. For this reason a large share of the cost of maintaining public schools should be borne by a local tax levied by the county or by the town in which the schools are. State aid is to be regarded as supplementary to, and not as a substitute for, local taxation for school purposes.?
On respect for legal procedures: ?Disregard for law and for its established modes of procedure is as serious a danger as can menace a democracy. The restraint of passion by respect for law is a distinguishing mask of civilized beings. To throw off that restraint ... is to revert to barbarism. It is the duty of the schools so to lay the foundation of character in the young that they will grow up with a reverence for the majesty of the law. ... A democracy which would endure must be as law abiding as it is liberty-loving.? (NEA, p. 284)
The cited Science papers from 1903 are rich in opinion and specific suggestions for how to teach as well as cautions about how to judge the effectiveness of teaching ? both topics still hotly debated today. For example:
On teaching mathematics: ?As a pure mathematician I hold ...that by emphasizing steadily the practical sides of mathematics ... it would be possible to give very young students a great body of the essential notions [of mathematics]. ... This is accomplished, on the one hand, by the increase of attention and comprehension obtained by connecting the abstract mathematics with subjects which are naturally of interest to the boy... and, on the other hand, by a diminuation of emphasis on the systematic and formal sides of the instruction in mathematics. Undoubtedly many mathematicians will feel that this decrease in emphasis will result in much, if not irreparable injury to the interests of mathematics. But I am inclined to think that ... under skillful guidance [the boy] will learn to be interested not merely in the achievements of the tools but in the theory of the tools themselves, and that thus he will ultimately have a feeling towards his mathematics extremely different from that which is now met with only too frequently ... a feeling that mathematics is merely a matter of symbols and arbitrary rules and conventions.? (E.H. Moore, p. 408)
On evaluating teaching: ?Character is the result of heredity and environment. To apportion the relative value of these influences in any case is no easy matter. If a school boy proves incorrigible it is generally attributed to heredity; if he becomes tractable, to environment ? so easily do we let ourselves be persuaded as to the beneficial effect of our influence. ... The current drift of educational thought is towards the perfection of methods and of systems of teaching. It is one of the happy signs of the times that teachers of all grades and all degrees of experience are trying to tell their brother and sister teachers how this and that subject should be taught. ... And yet these sincere and devoted souls, who have their daily reward in the bright and responsive faces of their pupils, generally overlook the fact that their success is not due so much to their methods as to themselves.? (Drown, p. 523-524)
As today the impact of tests is hotly debated ? from the school-level tests required by the ?No Child Left Behind? law to the college-prep AP and SAT tests mandated by competitive pressure of selective colleges ? so in 1903 were the influences of educational examinations:
On school exams: ?[In England] a committee was appointed ?to report upon improvements that might be effected in the teaching of mathematics...? One important purpose of the English agitation is to relieve the English secondary school teachers from the burden of a too precise examination system imposed by the great examining bodies.? (E.H. Moore, pp. 406-407)
On college entrance exams: ?Most of our larger eastern colleges still insist on their own entrance examinations. This makes a break in our education system which affects unfavorably the high school course preparatory to college, inasmuch as this course is then too apt to have for its aim the successful passing of examinations rather than a serious preparation for advanced work. This is an old and much-discussed question and I touch upon it now to assert my conviction that ? the diploma of graduation, accompanied by the personal statement of the principal, will become much better evidence of a boy?s fitness to enter on college work than a few days written examination can be. ... Whether or not he is likely to prove a diligent student with tastes and aptitude for his work, the college gets no indication from the examination papers...? (Drown, p. 522)
On admissions standards: ?I urge only that every [pupil] ... should have a motive for all he does. A motive, indeed, is more important than much knowledge, for it brings zeal, ambition, and earnestness, so often, so deplorably often, lacking in the college undergraduate. ... We are too careful as to the kind and amount of preparation a student has when he enters college and too careless of the work he does while in college. Some of the best and most successful students I have ever known have been those whom the college rules would have excluded, while many a one who fulfills all technical requirements has been a dismal failure.? (Williston, p. 134)
Today we are well aware of the many challenges associated with increasing numbers of students entering postsecondary education. In 1903 a similar expansion was taking place as the old classical curriculum fragmented into programs for students of different interests. Here are some implications, at least as seen through the eyes of one influential mathematician of the time:
On tracking: ?There remain some controverted questions ... which we cannot discuss without arousing prejudice which is attributed either to irrational conservatism on the one hand, or to sweeping iconoclasm, on the other. Even at the present day, many [hold] that studies may be divided into sharply defined categories designated as ?liberal,? ?humanistic,? scientific,? professional,? ?technical,? etc. ... They say, by implication at least, that mathematics, when pursued a little way, just far enough to make a student entertain egotistic but erroneous notions that he knows something of the subject, is an element of liberal training. On the other hand, if the student goes further and acquires a working knowledge of mathematics, his training is called professional or technical.? (R.S. Woodward, p. 163)
On intercollegiate athletics: ?There is a noisy minority who have succeeded, apparently, in convincing the public ... that one of the principal functions of an educational institution is the cultivation of muscle. ... There has sprung up, also, a class of less strenuous men who, taking advantage of the elective system, are pursuing courses of aimless discontinuity. ... They toil not, except to avoid hard labor; neither do they spin except yarns of small talk ... These types of men ... are now wielding an influence distinctly inimical to academic ideals ... Pray do not misunderstand me. ... The ancient maxim of sound mind in a sound body is more fitting now than every before. ... My protest is not against school and college athletics as such, but against athletics as they are now generally carried on, and especially against intercollegiate contests. As now practiced, athletics ... cultivate almost exclusively the men who are usually more in need of intellectual training.? (R.S. Woodward, pp. 166-167)
Finally, as a bracing example of the cultural assumptions of the era as well as of the primitive state of data analysis just one century ago, I offer this extended passage from a commentary that statistician Karl Pearson wrote in Biometrika about a previously published review of a memoir dealing with the homogeneity of the Naqada prehistoric crania.
On statistics: ?This is the material which the reviewer set himself to work out, taking the data haphazard from Flower?s well-known catalogue of skulls in the Royal College of Surgeons? Museum. ... The means of the variabilities of the skull lengths ... is 6.2788 and of the skull breadths is 5.0804. Mixing Australians, Guanches, Eskimos, and Chinese [the reviewer] finds a variability of skull length = 8.389 and of skull breadth = 7.002. He then points to the differences (2.1102 and 1.9216) and triumphantly asks how such small differences can be of any importance!
?But had [the reviewer] had a mathematical training he would know that nothing is ?small? absolutely, but only relatively to something else, and had he had a statistical training he would have known that he must compare it with the variability of these variabilities i.e., the standard deviation of the standard deviations of the skull measurements. Now the standard deviation of the above series of skull variabilities = .5185 and that of the skull breadth variabilities is .7996. The first ?small? difference is therefore 4.07 times its standard deviation, the second is 2.40 times its standard deviation.
?In other words, whatever sort of group the Naqada, Bavarian, Aino, French, and English male cranial series make, the odds are 42,552 to 1 against such an excess of variability as [the reviewer] found for his mixed series of skull lengths belonging to a number of that series, and 121 to 1 against such an excess as he found for the skill breadths occurring in such a series! It is such odds as these, the combination of which can hardly fall short of 4,000,000 to 1 and which no sane man in practical conduct could disregard, that amount to ?small? differences from the standpoint of the old school of craniologists!? (Pearson, p. 345-346)
Back to the Future
This passage shows in an explicit and sometimes startling way the nature of scientific thinking of this era, just one hundred years ago. In addition to demonstrating the roots of statistics in eugenics, it shows how undeveloped were scientist?s understandings of variability and standard deviation. If we are searching for evidence of progress, this may be a good place to look.
With this I return you to 2003, to judge for yourselves the progress we may or may not have made and to imagine, based on present progress, what may be said on this subject one century hence.
Lynn Arthur Steen is Professor of Mathematics and Special Assistant to the Provost at St. Olaf College in Northfield, Minnesota. He was President of the MAA in 1985?86.