to reproduce forms spoken or otherwise delivered or assigned by teachers. From primary school to college typical lesson assignments presuppose that lessons are to be looked at or heard and reproduced to the teacher in the way he wants them delivered to him. And, altho the recitative attitude signifies poor teaching and vague conceptions of the teacher's relation to the one taught, it is still the pedagog's mainstay, his stock in trade, his source of greatest pride. To lead young teachers to use it effectively and yet to realize its utter inadequacy by itself is one of the hardest and longest tasks in the preparation of all teachers.

3. The reiterative attitude is the recitative with concentration a little prolonged. It is based upon good receptivity. But the reciter in this attitude is unduly conscious of the forms of expression. He lacks spontaneity. When started on a paragraph or a page which he is to reiterate, he is like a boy coasting; it is disagreeable to be upset. He can't get another good start without returning to the point of departure. But I have visited many highschool teachers and college professors who rely chiefly upon the reiterative attitude and glow with enthusiasm when a poor parrot of a child can repeat, perchance in his own words, a long paragraph or a long lesson.

4. Without a generation of college professors who know good teaching and practice it, the preparation of high-school teachers can never succeed very well. So often the professor says to his students: "Read the book and get the author's thought;" or, "Listen to me and get my thought." But reading is not getting another person's thought. Reading is thinking; and hearing-language is thinking. So long as teachers and pupils meet chiefly for recitation their thinking is of a low type. Infinitely better than reciting and reiterating is cogitating. Every true teacher secures from each one taught the cogitative attitude of mind. But the typical professor dislikes to be interrupted in his lectures. He desires students to hear and reproduce "in substance" what he says. He seems not to know that hearing-language and observing and reading are all thinking processes requiring continuously the cogitative attitude of the mind. He is too commonly a recitationist; but he influences tremendously the high-school teachers. They follow his ways. His apparent purpose is to produce reciters rather than thinkers. He thinks and formulates for them. They recite after him. How delightful it is to run across those rare ones among us who are skilful in having students work out and think out and formulate subject-matter with them.

It is for normal schools and teachers' colleges to recast a great part of the current conception of the teacher's function and by a large variety of teaching experiments to bring all prospective teachers into a condition of constant eagerness to teach skilfully thru utilization of the ever-varying attitudes of those to be taught.

5-7. The inquisitive, skeptical, and critical attitudes of mind are suppressed in a large proportion of high-school and college classes. The typical recitation hearer does not enjoy them. They savor too much of disrespect for his

dogmatism. They throw him off his beaten track. They disturb his habit as a recitationist. They dislocate the adjustment of his oft-repeated story. They are too much like common life outside the school; they turn the mind from form to content. They lead toward definite questions, answers, arguments, and conclusions. They force issues to finalities. They are the delight of the full-fledged artist teacher in every school of every kind.

8. Another characteristic of good teaching is the combative or disputative mental attitude which implies living together as student and teacher and struggling with one another in friendly combat. In this attitude the student would not hurt the teacher's feelings, the teacher would not play boss or dogmatist, both student and teacher delight in courteously making unlooked-for interpretation of things, teacher and student live together in subjects, work out things together, indulge in sparkling, friendly cross fire, and welcome. witty retorts made in good temper. But how can normal schools and teachers' colleges prepare teachers to skilfully utilize this state of mind? Partly, perhaps, by instruction, but more by exemplifying it thru companionship with students in classrooms while teaching classes in the ordinary academic subjects. And the college professor should give us a square deal and do his share. 9. The discursive or argumentative attitude of the mind is better still. As a school inspector I many times longed to discover some difference of opinion between the high-school teacher and his students. The peaceful, monotonous harmony which commonly prevails in the high-school classes, means low mental vitality and wasted opportunities. It marks long and slow growth into habitual credulity. Where the critical, honestly skeptical, inquisitive, cogitative attitudes are utilized, the many persons taught see and think of many things which the one person who teaches cannot see or think of. Frank and honest exchange of ideas as to how things look does not mean waste. It means joint action and larger thought product. It means divided responsibilities and definite conclusions. It does not mean opinions formed by teacher and taught to students. It means conclusions that stick forever because they are worked out in the friendly competition of many persons, each one's notion being tested by the criticism of many others.

10. Best of all is the constructively synthetical attitude. It is seldom found in the typical high-school recitation. It is sometimes found in the grammar-school grades where alert, well-taught, masterful teachers dare allow their pupils to think for themselves, to struggle with subject-matter, to sum up or build up conclusions and declare where they are, how far they have come, and what they anticipate in view of the mental structures already erected.

This list of attitudes is illustrative, not exhaustive. The typical normal school delivers recipes and prescriptions for doing things. The teachers' college in the university is perhaps a little worse; it quotes from a larger bibliography. Both normal schools and teachers' colleges are consuming their best energies learning and reciting what some one has thought and formulated. But the poorest thing by which we deceive ourselves is the mechan

ism called the recitation. It assumes the student to be a reflecting machine to receive and return ideas and impressions. Professors who rely chiefly upon the lecture, the "quiz," and the "exam" seldom appreciate any process above the recitative. They assume receptivity. They are satisfied to receive back the content of talks and textbooks. When, by repression and bodily inaction, students lapse into somnolent torpidity, then inefficiency finds relief in notebooks. Voluminous copies of profoundly obscure lectures are kept. Bodily action in note-taking keeps awake the students of many an inefficient professor. There is fatal sequence. Stenographers copy into notebooks what speakers say, put aside notebooks feeling free from worry of cogitation, and later on reproduce from notes exactly what was uttered. In like manner the pedagog substitutes transmission for cogitation, obstructs thinking, prevents face to face contact with living teacher and snatches away opportunity to comprehend and assimilate subject-matter while fresh and new.

"Quiz" follows lecture, further disguising professional unfitness. "Quizzing" is not teaching. "Quizzing" narrows thinking of many into channels of one. The "exam" concludes the hampering process. Much lecturing and "quizzing" call for much examining because teacher is ignorant of student's mental content and attitude. But lecture, "quiz," and "exam" are the stock in trade of many a friend of ours who never dreams of cogitating, analyzing, questioning, arguing, and working out with students. the subject-matter to be dealt with, digested, and assimilated.

IV. CONCLUSION

All teachers during their professional preparation need in common: 1. To secure by instruction and experience a working knowledge of childhood and adolescence.

2. To acquire in teaching the habit of basing daily instruction on the learner's mental content and attitude in order to modify both his content and his attitude and accustom him to the habitual and independent reorganization of his mental content.

3. By trial in many phases of experimental teaching they need severally to discover themselves and what their several talents are, and in view of their talents inherited and acquired, what they are severally destined to do best.

To do all this will consume by far the greater part of the time and energy which teachers can devote to initial preparation.

Probably one-tenth of the labor in the professional preparation of teachers should be devoted to special pedagogical aspects of subjects to be taught. In these special aspects high-school teachers and elementary teachers, after differentiation and near the end of their professional preparation need separate instruction in such things as bibliographies, appliances, and the correlation of each separate subject with other parts of the curriculum.

ROUND TABLE CONFERENCES

B. MATHEMATICS ROUND TABLE

ADAPTATION IN MATHEMATICS

CHARLES AMMERMAN, HEAD OF DEPARTMENT OF MATHEMATICS, WILLIAM MCKINLEY HIGH SCHOOL, ST. LOUIS, MO.

matter.

The problem that is constantly before the teacher of mathematics is the adaptation of the subject-matter to the pupil. Commonplace as the statement may seem, the problem is a comparatively new one. The effort has always been to adapt the pupil to the subjectScholars in geometry and algebra have made certain organizations of those subjects. The great mass of teachers has taken these organizations almost without question, placed them before their pupils, and insisted on their getting them, usually requiring all to get them in the same way. If this "same way" did not happen to suit some members of the class, the teacher complacently assumed that they had no mathematical knowledge or power, and graded them failures. Except where classes are small, and it is easily done, little or no allowance is made for individual mental makeup, previous training, or immediate environment. Pupils are herded, and the work is adapted in a general way to the herd. One of the greatest questions before the teachers of mathematics is, how can two pupils in a same class be given mathematics when one requires an entirely different procedure from the other—that is, how the subject can be presented in a way adapted to the learning mind.

There is no subject in the high school in which so many pupils fail as in mathematics, which is only another way of saying there is no subject in the high school which is so imperfectly adapted to the pupils. You may examine the records of any high school you please, and you will find that where there is one failure in the languages or the sciences, the chances are there are four or five, or possibly more in algebra or geometry. I do not believe this should be so. If a body of pupils has met the standard required for admission to a class, it is doubtful if 10 per cent. of them should fail. Many would make the per cent. smaller, yet 20, 30, and even 50 and 75 per cent. are sometimes compelled by the teacher to repeat the work. I have in mind now a school which is typical in this respect. One of the teachers received pupils who had been given to understand that they had completed the work satisfactorily in the preceding classes. The teacher, when I knew him, had gone over the work for a number of years. He had the outline and the details clearly in mind. He utterly lost sight of the fact that the pupils in the class had not been repeating this subject for ten years, and that their experiences were very different from his own. So with the keenness and the sarcasm that too often comes out of such a condition he usually told about 40 per cent. of his class that they knew nothing about the subject, and that they would have to take it over, tho it was doubtful if that would do any good. This same teacher, as I remember it, took pride in the number that failed to do the work in his classes, and was inclined to boast of it. While the case I have described is extreme, it is not uncommon. I believe it is true in some degree in most of our high schools. The whole thing may be summed up in the statement that the mathematics of any year should not be harder than the pupils of that year can get, or presented in a way that the pupils cannot grasp. If a large per cent. of failures exist, it is more of a reflection on the teachers' ability to adapt than on the pupils' ability to master. Teachers who take pride in the fact that they make a large number of their pupils fail, or that they make it difficult, if not almost

impossible, for their pupils to pass in their work, will not be likely to look upon such a statement with favor.

To make more definite what I mean in the sense of adaption, consider the subject of algebra. While thinking about this question, I picked up one of the algebras on my desk, which is not especially different from most of the algebras in use, and found that the first eighteen pages were devoted to definitions, and the next twenty-five or thirty to abstract work. The definitions were generalizations, that are usually beyond the experience of the pupil; the abstract work, or exercises, as they are usually called, go smoothly enough, for if he sees a few of them work, he can easily get the others. These abstractions do not mean anything particularly; they do not stand for anything. • After he has gained some mechanical expertness, he is given some applications, and every teacher of algebra knows how difficult it is to have him grasp those applications. Ofttimes they have nothing whatever to do with his experience. Just now it is the proper thing to use problems in the field of physics, which is helpful, provided, as usually happens, the physics involved is not more obscure than the mathematical principle. A year or a year and a half of such work, which is largely abstract, is given, followed by about the same amount of geometry. A majority of high-school students do not get mathematics during the senior year, except as they find it in some science, as physics. Then comes the college work, and many find they have forgotten all they knew.

Last Thanksgiving, the Central Association of Science and Mathematics met in Chicago, and an instructive paper was read before the mathematics section on the mathematics of the entering students of the Wisconsin University. The writer stated that a very large per cent. of the candidates made a very poor showing. He attributed this to the fact that too many of them had had poor teachers in the subject when they were in the high school. I do not recall that he said any thing about an equal failure in the languages or sciences; it seems that the trouble was in mathematics mainly. The examination had ten questions, eight of which were abstract and two concrete. The student had been expected to recall something he had not had for some years and which had no meaning to him when it was first studied. Is it so surprising that so many pupils know so little of mathematics, when they enter college or better when they leave the high school?

I doubt if any abstract work should be given to an algebra pupil during his first year; certainly none should be given during the first part of that year. Whatever is not concrete to him might as well be left out. I may work industriously on the definitions and abstractions usually given during the first lessons of algebra, and the pupil will not have as clear conception as he would if I said nothing about them, and got at once into the heart of the work with a problem something like the following;

The shortest railway route from Chicago to New York is 912 miles; how long does it take a train averaging 38 miles an hour to make the journey?

Solution in words: The product of the average number of miles per hour and the required number of hours equals the whole distance traveled; that is, 38 multiplied by the required number of hours equals 912. Hence, the required number of hours is one thirty-eighth of 912, or 24.

Solution using abbreviations: If, instead of the expression "the required number of hours," we use the word "time," or simply the abbreviation "T," the solution may be written: 38 x T=912. Hence T =912 divided by 38=24.

The knowledge that the pupil has when he begins a subject should be made the basis of his work, rather than the knowledge of some learned man who made problems in physics or some other subject, which the pupil knows nothing about. The teacher of the past was satisfied to deal with the logic of the subject; the teacher of today has a harder task before him. As yet, most of us are satisfied to put the text into the hands of the pupil and accept results. Perhaps we are satisfied because it is the easier thing to do. Many do not like to do otherwise. Many will not do otherwise, but the future demands a change.