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complex, and in many cases indivisible into units. The quantity, a fifth-grade boy's ability in addition, is not a constant, but varies from pupil to pupil. So also the ability of a group of pupils of a given grade varies from class to class; and in the same way morals vary from community to community. The salaries paid to teachers, the amount expended on education per pupil in cities, the ratio of a city's expenses for education to its total expenditures, etc., are all variable within the group. Therefore, the complete measure of any intellectual or moral quality in a group is not any single figure, but a list of figures, each representing one manifestation of the quality's amount, and all together representing all its manifestations. The adequate measure is so much of this list as shall give as precise a representation of its total make-up as our purpose requires. Now all the habits of school arithmetic and of account-keeping with dollars and cents impel one to treat all quantities as constants. Moreover, not one man in a hundred, even of college graduates, is familiar with the arithmetic of variable quantities. Yet presumably knowledge of it is the first step toward the successful quantitative study of education. At all events, the measurer of mental facts who ignores their variability can handle only the cruder problems and attain only insecure results.

The complexity of the phenomena of education is obvious and need not be discussed here. The lack of commensurate units in which to measure intellectual and moral qualities has been by far the greatest barrier to quantitative science, all the more so in education because we have been unconscious of the lack. The units at hand are defective in that they are subjective, depend on individual caprice instead of universal agreement, are not commensurate, and are reckoned from unknown. zero-points. In even so simple a matter as ability in arithmetic, the measure to be given to any individual is not an easy affair. In the first place, the marks will vary somewhat with the individual who examines the examples done; secondly, the credit to be given for any example as compared with any other is unknown; and, next, even if the test comprised twenty examples known to be of equal difficulty and to have been marked impartially, the fact that one boy scored 14 and another 7 need not mean that the former is twice as far above zero in arithmetical ability as the latter.

This indefiniteness of the zero points of intellectual and moral scales deserves more attention than can be given to it here. In physical science we can reckon distance, weight, velocity, acceleration, energy, and the like from absolute zero. Zero length, zero mass are quantities as

definite as any others. Two pounds is two times one pound, not only in the sense that it is made up by the addition of two equal units, but also in the sense that it is twice as much more than zero as one pound is. But, while the doing of one set of seven examples may involve the same arithmetical ability as the doing of another set of seven, doing fourteen need in no way involve an ability twice as much more than zero ability in arithmetic as is the ability to do seven. For the real zero of ability in arithmetic may be far below that which just fails to do even one example. In fact, the zero point for arithmetical ability is unknown; and if one chose any point as it, he would have only such an arbitrary point on the scale as the 0 Fahrenheit or the 0 Centigrade of temperature. At the present time even such arbitrary zero points have not been fixed. The result is that, as I have elsewhere stated,* researches in child-study are full of such arithmetic as this:

John, who weighed 4 lbs. more than 100 lbs., has added 2 lbs. to his weight; James, who weighed 100 lbs. more than 10 lbs., has added to his weight 50 lbs. Both gained 50 per cent, and so their relative gains were equal.

John weighs 10 lbs. more than 60 lbs. James weighs 2 lbs. more than 60 lbs. John is five times as heavy as James.

The measurement of things or existences, however, is child's play compared to the measurement of changes, and I defy any one to answer the following apparently simple question: John, being tested with 500 words, spelled 200 correctly; a year later, after a certain training, he spelled 300 correctly. Fred, under the same conditions, spelled correctly 400 and 500. Who made the greater gain in spelling ability?

In the first place, to spell one word correctly is not the same quantity as to spell some other word correctly. But even if the words were of equal difficulty, the question must remain unanswered. He who claims that, since a hundred things are equal to another hundred of the same kind, the gains were equal, must claim that the child who from the inability to spell any word comes to spell 100 gains no more in spelling ability than the boy who, knowing how to spell 2,000, learns 100 more; that the child who from total idiocy comes to use a dozen words gains no more in intellectual ability than the linguist who adds a dozen foreign words to his vocabulary.

On the other hand, whoever claims that since John increased his measure fifty per cent, and Fred only twenty-five per cent, John gained twice as much, must agree that a bright infant learning its third word

*"An Introduction to the Theory of Mental and Social Measurements," p. 16.

makes as great a gain in speech as a master of English, Latin, French, and Gaelic who learns, say, Russian and Chinese.

The fact is that the application of ordinary modes of arithmetical thought to the problems of change in mental or moral abilities is beset with pitfalls, and that unless these pitfalls are kept constantly in view, the most elaborate investigations will go astray.

Finally, the measurement of the relationships or dependencies of intellectual and moral facts is so complicated by their variability and the indefiniteness of the zero points for the quantities in question, that its problems cannot even be understood without a knowledge of the theory of measurement of variable quantities. Yet it is precisely measurements of relationships that are needed for the solution of educational problems. The first step toward the discovery of how far heredity determines human achievement consists in the measurement of the dependence of the ability of a child upon that of his parents. The influence of any environmental agency will be most exactly known by measurement of the relation between varying amounts of it and the corresponding amounts of the quality it is supposed to increase. The causes of efficiency in any intellectual or moral trait may best be discovered by measuring the relationship of that trait to each of its possible causes.

I trust that enough has been said to convince my readers that the measurement of the phenomena of school life makes severer demands than weighing coal or keeping a cash account, and that the equipment of the scientific student of education should include at least an elementary acquaintance with the theory of measurement of variable quantities. It is folly to expect to gain the reward of scientific method without assuming its responsibilities. It is not enough to improve on casual observation and imaginative hypotheses, though that is a very great improvement. To have in education the real benefits of quantitative science, we must spend arduous years in devising, testing, and standardizing units of measurement, in searching for convenient arbitrary zeropoints, and eventually for real zero-points, and in determining the sources and amounts of the errors of measurement. It has taken many centuries and the labors of many gifted men to develop the present system of physical measurements, and the task is far harder in the case of intellectual and moral facts.

That the adequate measurement of intellectual and moral facts is difficult does not, however, at all imply that it is impossible or that inadequate and erroneous measurements are not useful. The eminent men who from time to time deny the possibility of mental measurements

may be divided into three classes: (1) Those who are ignorant of the facts altogether and who would never have attained eminence by any power of comprehending the meaning of measurement; (2) those who mean by measurement the ordinary grocery store weighing and figuring, and are impressed by its inadequacy for mental life; and (3) those who mean by intellect and morals an unapproachable inner life which we can only be, and not know in the ordinary sense. What the world at large means by intellect and character is measured whenever we apply any comparative adjective to a human being. Even of human emotions, the most qualitative of mental facts, measurements are every day made. Love, envy, and joy are greater or less as truly as are feet and inches.

Inadequacy and error are relative terms. It is a great advance to regard the sun as ten times farther away from the earth than the moon is, if the previous belief was that it was only half as far. The measurements of ability in composition made by Dr. Rice, for instance, are surely subject to errors, but they are infinitely better than the guesses of the customary theorizing on the subject. Only the ignoramus regards any measurement as perfect. All are approximations, and a very rough approximation may be far more accurate than previous measurements.

Nor should the peculiar difficulties of quantitative work in education discourage any one from trying to do it. Teachers and administrative officers everywhere will profit from making systematic observations and keeping precise records of school work. They will be less misled by quantitative studies than by guessing. Of course, we shall not advance the knowledge of education by leaps and bounds, for the same reason that would prevent us from so advancing physics or astronomy or chemistry or biology. It requires great natural gifts, special training, and extreme devotion to make great progress in any field of knowledge. sane person should expect that any change in habits of thought will transform the men and women engaged in educational work into a group of scientific experts. Nor does quantitative work need any such expectation as its justification. All that is asserted is that it is a useful tool for the thinker and is commonly better than the work it replaces.

No

The men who are responsible for large educational enterprises, who control the expenditure of hundreds of thousands of dollars annually, can well afford the expense of expert quantitative study of the work of their schools. It is cheaper in the end to have one man who knows the theory and technique of mental measurements than to have three who do not. It is, in fact, cheaper in the end to get expert advice whenever it is to be gotten. The superintendent of schools who can add to his

present dreary annual recital of bills paid, attendances, tardinesses, buildings, janitors, and the like, a definite, comprehensible, and reasonably exact statement of what the boys and girls have actually gotten from that year's schooling, can afford to pay well for it. Few competent business men would hesitate to pay one-tenth of one per cent of their operating expenses for expert advice concerning production, advertising, account-keeping, and the like. But even that small proportion would mean some $20,000 annually for the scientific study of school work in New York City. No one can doubt that it would pay, if the expert service were obtainable.

It is obtainable. Young men are being so trained to-day as to possess methods of investigating and treating the facts of education which were unknown to the students of a decade past, and which are as far in advance of the scientific equipment of the present generation of school officers as the methods of the expert chemist are in advance of those of the manager of a foundry, refinery, or mill. The managers of school systems, like the managers of business enterprises, political parties, or manufactories, must for a long time yet, if not forever, be primarily gifted in dealing with men, money, and opinions. They cannot themselves be exclusively, and can rarely be to any great degree, investigators and statisticians. But, like the managers of business enterprises, they can appreciate and use expert investigators and statisticians. That they

have not done this hitherto is due not to any failure on their part or on the part of school boards to conduct their business as efficiently as steel foundries and mining companies conduct theirs, but to the non-existence of the expert. The supply will, in this case, as in the case of the technical industries, assist in creating the demand.

The exact study of the business of educating children is useful. Therefore it will soon be used. No one need fear that the seed now being sown by men who study the theory of measurement of the facts of education, and who apply the proper methods to the investigation of school work, will fall on barren ground. The educational practice of this country is eager to profit by any aid which science can give. I do not mean that superintendents of schools are as enthusiastic about exact scientific methods in education as I am. It is not their business to be, and it is my business to be. They have many other things to do, and I have not. But I do assert that no other business will welcome more

heartily and reward more fairly the scientific expert than will the busi

ness of administering schools.

EDWARD L. THORNDIKE.

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