weeks, but 5 schools extend it through a full year of 34 or more weeks. Solid geometry and trigonometry are always reported as half-year courses.

Periods per week.-The number of periods per week in elementary algebra, with one exception each of four and six periods, is uniformly five. The exceptions to this standard practice in plane geometry are only three, two of four periods and one of six periods. The exceptions in advanced algebra, solid geometry, and trigonometry are the same, one each of three and four periods.

Length of periods.-The lengths of periods are shown in Table XXV. With a small proportion of exceptions the lengths of periods

TABLE XXV

NUMBER OF SCHOOLS REPORTING VARIOUS LENGTHS OF CLASS PERIODS FOR COURSES IN MATHEMATICS

are 40 and 45 minutes, the practice being almost evenly divided between these two. The one school each under elementary algebra and plane geometry reporting 80-minute periods devotes half this time to supervised study. Both schools reporting 65-minute periods in elementary algebra and plane geometry report supervised study during 30 minutes of this time, the remainder of the time being given over to recitation. Both of these schools require outside study in addition. Three other schools reporting 40-minute periods for elementary algebra and one school each reporting

40- and 45-minute periods for plane geometry provide additional periods of equal length for supervised study.

THE REQUIREMENT IN MATHEMATICS

Statements were received from 106 schools as to the amount of work in mathematics required for graduation. The facts are presented in Table XXVI. The table makes clear the following facts: (1) that the usual requirement of mathematics is two years; (2) that a small proportion of schools require less, some even having dropped mathematics as a required high-school subject; (3) that some schools still require two and one-half or three years, some of the latter perhaps out of deference to the older and now disappearing

TABLE XXVI

NUMBER OF YEARS OF MATHEMATICS REQUIRED FOR
GRADUATION FROM THE HIGH SCHOOLS

college-entrance requirement of three years of mathematics; (4) that a single school imposes a four-year requirement; and (5) that 2 schools state that the requirement depends upon the course pursued by the student. Certain qualifications added by a few of the teachers indicate that a practice similar to that just stated under (5) appears in more schools than the tabulation enumerates. One school reporting a requirement of two years in most courses specifies three years for technical courses. Another school adds a third year to the usual two-year requirement for those taking the college-preparatory course. A few other schools permit departure from their usual two-year requirement in "some courses," but do not name these courses.

III. ORGANIZATION OF THE COURSES

The answers to the question, "What important deviations do you make in your course from the plan of the text you are using?" give full support to the conclusion that the content and organization of courses in mathematics are largely determined by the textbook used. It will be noted in Table XXVII that a large proportion of the teachers report that they make no important deviations. To these, because of the conscientious way in which the teachers generally have responded to our inquiry, we may safely add practically all of those who make no answer to the question.

TABLE XXVII*

DEVIATIONS FROM THE PLANS OF TEXTS USED REPORTED BY TEACHERS

* Because a few deviations do not conform to the classifications adopted here, and also because a few teachers report two or three types of deviations, the numbers under the various categories are not, except in two instances, equal to the total numbers of responses to the questionnaire. These last have been introduced merely for purposes of comparison.

Almost all deviations reported were readily classifiable under the categories "Omissions," "Additions," and "Shifts of Order" appearing in the table. This may be illustrated for all the divisions of the field by quotations from typical deviations reported by teachers of elementary algebra: "omit some theory," "add drill work," "shift order," "defer graphing," "omit graphing," "simpler problems added," "factoring before fractions," "much extra work," "give mimeographed lessons" in addition, "introduce transposition early," etc. Only a few report deviations of as much significance as "commence with equation and make all else subordinate to it" and "correlate various branches of mathematics." In the case

of plane geometry 10 of the 48 "additions" reported refer to the introduction of "practical" problems.

The teachers were asked to state what fractional parts of the recitation periods are devoted to recitation, study, teaching, and lesson assignment. Although the practices were ascertained for all the divisions of the field of mathematics, because these do not vary from subject to subject in any significant respects, only those

* The numbers in parentheses are the numbers of schools reporting the practices.

reported for the teaching of elementary algebra are reproduced here (Table XXVIII). From one-eighth to three-fourths of the class period is devoted to recitation proper, the modal practices being one-fourth, one-third, one-half, and two-thirds, these modal points suggesting a wide range of practice. From none to one-half of the class period is devoted to study, the modal points being none, one-fourth, and one-third. The noteworthy facts here are that a very large proportion of schools allow no time for study and that those who provide it restrict it to a small proportion of the class period. Teaching occupies from one-tenth to two-thirds of the period, the modal practices being two-ninths to one-third. Lesson assignment occupies one-twentieth to one-half the time, with modal practices at one-ninth, one-eighth, and one-fourth. It may be said that the data on this matter lack reliability in some degree because of the possibility of different interpretations of the words "recitation," "study," "teaching," and "assignment." Teaching, in particular, suffers from such lack of uniform definition.

In connection with this discussion of the disposition of the class period, we mention again the practice in several schools, reported under "Time Element" above, of providing in classes in elementary algebra and plane geometry for supervised study in connection with, and in addition to, the regular recitation period of 40 or 45 minutes.

TYPES OF METHOD FOUND MOST SATISFACTORY

The answers to the question as to which of the various methods, i.e., authoritative, deductive, inductive, analytic, and genetic, are being used have been assembled in Table XXIX. The deductive, inductive, and analytic methods seem to be most used, the authoritative and genetic being used by only a small proportion of teachers.

TABLE XXIX

TYPES OF METHOD FOUND MOST SATISFACTORY BY TEACHERS OF MATHEMATICS

Many teachers report the use of more than one method, sometimes three or four, some stating specifically that different methods are pertinent at various times. Several other methods or other names for those already listed are reported by one or two teachers each: "grouping theorems," "heuristic," "lecture and dialogue," "synthetic," "development," and "suggestive."

SPECIAL DEVICES

A

Table XXX shows the extent to which certain special devices are being used by the teachers of algebra and trigonometry. small number of teachers report that they are using tables of