squares, cubes, and cube roots. Only a few teachers of geometry report the use of the devices named in the table.

TABLE XXX

EXTENT TO WHICH CERTAIN SPECIAL DEVICES ARE REPORTED AS BEING USED IN TEACHING ALGEBRA AND TRIGONOMETRY

The following percentages of the teachers report the use of historical notes in their classes in mathematics: in elementary algebra, 37.8 per cent; in solid geometry, 30 per cent; and in trigonometry, 36.4 per cent. The small proportion of teachers who report the use of "a very few" and a number of others who make no answer to the question may be included with an approximate third of the teachers who answer "no." Some of the teachers who report that historical notes are not introduced plead lack of time as the reason for not making them a constituent of courses in mathematics. The figures just given indicate that such notes are more commonly used in the earlier than in the more advanced courses.

The values most commonly ascribed to this use of historical notes are the interest they add in "humanizing" the work and the light they throw on the development of the subject. On the other hand, some teachers who report that such notes are not constituents of their courses regard the historical aspect as non-essential, impractical, and unsuited to the maturity of the students.

CORRELATION OF ALGEBRA AND GEOMETRY

Table XXXI will give some information as to the proportion of teachers who are making efforts to correlate the work in algebra and geometry, although it can offer little as to the extent of correlation within the schools. In general, we may say that those answering

"yes" or "some" may be considered as having done something in the way of such correlation, whereas those who answer "very little," "no," or who do not answer may be classed together as

TABLE XXXI

EXTENT TO WHICH TEACHERS REPORTING MAKE EFFORTS TO CORRELATE ALGEBRA AND GEOMETRY

doing nothing or practically nothing along this line. In figures this will mean that a total of 174 teachers have done something and that 107 have done nothing or practically nothing with such correlation.

The values which teachers say they have found in such correlation classify, with very few exceptions, under the following: (1) it makes the subjects easier of comprehension; (2) it teaches the unity of mathematics; and (3) it creates more interest in the subjects. A few teachers complain that such correlation confuses algebra students and that therefore the attempt in their schools has not met with success.

EFFORTS TO MEET CURRENT CRITICISMS OF HIGH-SCHOOL MATHEMATICS

The teachers were asked to describe briefly any efforts they have made to meet current criticisms of high-school mathematics. Table XXXII indicates that 174, or approximately 60 per cent, of the questionnaires signify that the teachers are making some effort to meet such criticisms. Those who do not answer the question may safely be included with those who report they are making none, thus leaving us free to conclude that approximately 40 per cent are making no such efforts. Perhaps many of these are of a mind with the one teacher who reports that he "ignores" the criticisms.

The kinds of effort described vary widely, and one is led to wonder how the teachers who report some of them can delude themselves into believing that they are seriously addressed to meet any criticism. For the most part, however, the efforts include reforms in the interest of the practical, the utilitarian. Under this teaching may be grouped efforts to use "problems of daily life,"

TABLE XXXII

NUMBER OF TEACHERS REPORTING EFFORTS TO MEET CURRENT CRITICISMS OF HIGH-SCHOOL MATHEMATICS

"more concrete work," "more practical problems," "vocational problems," and "problems within the students' experiences. For example, such are mentioned 45 times by the 66 teachers of elementary algebra who signify that they are making efforts to meet current criticisms. The proportion reporting this type of effort is approximately the same for the other divisions of the field of mathematics.

V. AIMS AND VALUES

AIMS

The aims and purposes of the work in mathematics were set down with a great variety of expression. They are reducible, however, to about four general classes. The classes call for (1) a working knowledge of the subject, (2) preparation for subsequent academic work, (3) a stressing of the practical aspects, and (4) free play for the disciplinary values. The extent to which the aims as classified are reported by the teachers is shown in Table XXXIII. In the minds of the teachers disciplinary aims are manifestly most

important. It should be noted that emphasis upon the practical aspects and allowance for the free play of disciplinary values are not considered mutually exclusive, some teachers reporting both.

TABLE XXXIII

EXTENT TO WHICH CERTAIN TYPES OF AIMS ARE REPORTED BY TEACHERS OF MATHEMATICS

That some idea of the method used in the classification of aims may be given, typical aims and purposes reported by the teachers of elementary algebra are quoted here. Under the head of "A Working Knowledge of the Subject" were placed such statements as, "to manipulate formulae and generalize,” “accuracy and skill in handling algebraic symbols," "easier solution for problems," "knowledge of the equation," and "to impart mathematical elements"; under "Preparation for Subsequent Academic Work" were placed such statements as the following: "aid in later mathematics and science," "to make the work in geometry easy," and "to prepare for college"; under "Stressing the Practical Aspects": "to make the work practical" and "to supplement and add to practical computation value of arithmetic"; under "Free Play for Disciplinary Values": "to develop power and accuracy," "to teach the child to think," "training the mind," "emphasis upon the disciplinary value,”" "to lay a foundation for analytic work of all kinds," "to teach definiteness, leading to assurance and poise," and "to develop the habit of taking initiative in any problem or

task." One cannot refrain from noting, on reading the statements in the questionnaires made in response to this inquiry, how much more glibly teachers speak of these disciplinary values than of those aims that classify under the three preceding heads.

EXTENT TO WHICH AIMS ARE FULFILLED

The teachers were asked to state to what extent they believe that their purposes are accomplished with various groups of pupils in the high school. The following groups were listed in the questionnaire: boys, girls, those preparing for college, and those preparing for trades. Nothing of significance appears when the answers are tabulated with respect to the four classes of aims listed above, as the teachers seem to have understood the question to be to what extent the courses in mathematics meet the needs of these groups of students. The proportions are the same for all four classes of aims. The tabulation of responses on the latter basis indicates that courses in mathematics more nearly approximate the needs of boys than of girls and of those preparing for college than of those preparing for trades. This is shown in Table XXXIV, which presents the facts for plane geometry and which will give a fair representation of the situation for the other divisions of mathematics as well.

TABLE XXXIV

PROPORTION OF TEACHERS WHO BELIEVE THAT
NEEDS OF VARIOUS GROUPS OF STUDENTS ARE
BEING WELL CARED FOR BY COURSES IN
PLANE GEOMETRY

The answers to the question, "Which do you regard as more important, the content of the course or the discipline?" appear in Table XXXV. The facts set forth here, as well as those presented under "Aims" above, go to show that, despite the recent