in school or elsewhere in getting a true appraisal of the worth to civilization of the performances of the men who bore those honored names? To most of us it seems a modest enough return for posterity to attach the discoverer's name to some important truth. Some of the appended names may not be musical to modern ears, but the worth of the services rendered by the bearers to subsequent investigation and discovery condones the want of harmony. Those who know the underlying facts see nothing offensive or in bad taste in Cartesian geometry, Tchirnhausen's transformation, or the theorems of Apollonius, Euler, Bernouilli, Newton, Wallis and a score or so others.

Underneath the movement for a greater use of the history of mathematics in the teaching of its several branches is the thought of emphasizing more the appreciation phase of mathematical learning. We have not yet gone as far as we ought in the use of historical and environmental backgrounds of mathematics but we are doing better year after year in the pedagogical evaluation. in the history of mathematics. The teacher who will give one or two class exercises a month to mathematical appreciation will find the mathematical spirit of the school very markedly improved in a surprisingly short time. The next best thing to a knowledge of mathematics is the right appreciation of its services.

Criteria for judging an appreciation exercise are as follows:

1. What agencies were being used to furnish appreciation of mathematical culture?

2. Had enough mathematical information already been given to enable pupils to appreciate the role mathematics had played in the situation? 3. If history was used, how was it used?

4. If modern uses in industry were employed, what were they?

5. Did the class seem to be making more intelligent their appreciation of mathematical study?

6. Did there seem any lack of appreciation arising from ignorance of mathematical knowledge or technic?

7.

Suggest any other means than those used, for aiding mathematical appreciation in a high school class.

8.

done?

Was the age of the pupils appropriate for the sort of thing being

9. Any difference between boys and girls in expressions of appreciation? 10. Would you favor using class exercises for appreciational work in mathematics?

11. Suggest ways of improving what you saw attempted.

12. Do you believe the time might better have been used in studying subject matter?

In concluding this discussion it may be said that professional expertness in the teaching of high school mathematics consists largely in first recognizing the type of thinking the particular

problem or topic calls for and then choosing the teaching technic that is most appropriate to the type of thinking in hand. This is real lesson-planning. An exercise in associational thinking should not be conducted in the same way as an appreciation, or a drill, or a problem-solving, or a concept-forming exercise should be conducted. Each type has its appropriate technic, and, in studying the teaching of others, the question is how does the particular sample exercise being judged compare with good work of its type? Because this is true it happens that some teachers are excellent in some class exercises and very poor in others. The point to be emphasized is that in both the planning and the judging of teaching discrimination of types of work is an essential condition to success. This is no less true because a given class. exercise may exemplify two or more of the types above mentioned. Finally, real benefit from the observational study of the teaching of others comes largely, if not mainly from the comparative judging of types of mathematical thinking involved and of the degree of appropriateness of the class procedure to the type of thought.

REQUIRED MATHEMATICS IN THE FOUR YEAR HIGH SCHOOL.

BY R. D. SHOUSE,

Principal High School, Kirksville, Mo.

ALGEBRA IS NOT A SUITABLE REQUIRED SUBJECT.

Few men have argued that the present organization of high school mathematics is not the best for pupils who aim to take college work which requires considerable mathematics. An increasing number of public school administrators and mathematics teachers are, however, coming to the opinion that to require all pupils to study algebra is a mistake. If the high school is to be primarily a college preparatory school, then the present organization is justifiable. If the high school is the people's college, with the minor purpose of preparing a few of its pupils for higher education, then the first year of the present organization is clearly unjustifiable as required work.

On the other hand there has always been a general demand for more mathematics than is obtained in the first eight grades. This demand is responsible for the high school mathematics requirements. A few schools have removed all such requirements and many will probably follow their example unless the first year course is adapted to the purposes of the public high school.

The movement toward the reorganization of the high schoo mathematics courses is well under way and it is the purpose of this article to discuss the new required course.

CONTENT OF THE NEW REQUIRED COURSE.

The consideration which should determine the content of the first course is its usefulness to the average first year pupil. The course should be practical, probably not entirely in the sense of bread and butter acquisition, but certainly it should not be an attempt to anticipate a pupil's school needs beyond the twelfth grade. In determining the content these questions should arise; first, "What will most of the pupils desire and need to know and do in the line of mathematics?"; second, "What is there in the field of mathematics which will best satisfy these needs that can be taught in a year?" Community interests and children's interest should be vital forces in determining the content.

A review of late texts on general mathematics for the ninth grade, of junior high school texts and of reports of committees on the reorganization of high school mathematics will show that the leaders of the movement are fairly well agreed as to the content of the first year course. It includes, briefly: Algebra, with emphasis upon the idea of relationship, the formula, variation and graphical representation of data; factual and constructive geometry, with emphasis upon congruent and similar figures and numerical trigonometry in the solution of right triangles by use of the sine, cosine, and tangent. In the matter of omissions from the course in algebra the recommendations of the National Committee on Mathematical Requirements are being generally followed. Portions of their recommendations follow. "Drill in algebraic manipulation should be limited to those processes and to the degree of complexity required for a thorough understanding of principles and for probable applications, either in common life or subsequent mathematics." "In addition to the large amount of drill in algebraic technique already referred to, the following topics should, in accordance with our basic principles, be excluded from the work of the first two years: Highest common factor and lowest common multiple, except the simplest cases involved in the addition of simple fractions; the theorems on proportion relating to alternation, inversion, composition and division; literal equations, except such as appear in common formulas, such as may be necessary in the derivation of formulas, the discussion of geometric facts, or to show how needless computation may be avoided: radicals, except as indicated in a

previous section; extraction of the square root of polynomials; cube root; theory of exponents; simultaneous equations in more than two unknowns; pairs of simultaneous quadratic equations; etc." The tendency seems to be away from attempting any demonstrative geometry whatever.

However, the first year course must not be characterized alone by the mere omission of certain topics but rather by a new treatment of the old topics. The content proper is children's needs met-let the degree of complexity, amount of drill and topics used fall where they will.

ORDER OF PRESENTATION OF TOPICS.

While leaders are fairly well agreed as to the mathematical topics to be taught, they differ widely in the order and mode of presenting these topics. In the formal presentation the order is determined largely by the subject matter itself. Logical development is the prime requisite. In the general course the development should be psychological. The subject matter must be grouped around problems of the children with little concern as to whether it is algebra, geometry or trigonometry. The determination of the best problems to motivate or project the work will require much experimentation. We want genuine children's interests and adults cannot casually assume them. While the present divergence of opinion indicates that we are still somewhat up in the air, it is a healthy sign and a prediction of a real accomplishment.

Our great failing is to become enthusiastic over a new idea and sacrifice its spirit to a superficial form. Just as the social recitation often degenerates into the usual recitation conducted by pupils, so is general mathematics apt to become a portion of algebra followed by a portion of geometry and so on. A mere stratification of algebra, geometry, and trigonometry is not as efficient, nor is it as satisfactory to the teacher or pupil as our present organization. Why? Because it is unquestionably a considerable satisfaction for many pupils to follow the logical development of algebra or geometry. There is satisfaction in having completed a well rounded task In a stratified course the class has no sooner acquired an interest or momentum in algebra than there is an abrupt change to geometry with different problems and different ways of thinking. The teacher gets a sort of jerky sensation and the pupils seem to be continually in hot water, never knowing where they are or what it is all about. If mathematics is to be taught in stratas, it is not advisable to de

vote less than about one-half of a school year to a subject. Our hope to make general mathematics a living, active subject depends upon our ability to take the methods of algebra, geometry, and trigonometry out of their several shells and apply them to the solutions of problems which may be made real to children.

TEACHERS AND GENERAL MATHEMATICS.

Before a large measure of success can be obtained teachers must be trained to teach the new course. There is no subject in the high school course of study as easy to teach as algebra, with the aid of one of the standard texts. Teachers have learned mathematics in the logical way and it is natural for them to teach mathematics in the same way. Any one of the present texts in reorganized mathematics, if used by a teacher who is accustomed to the use of texts in algebra and geometry, is apt to be reported a failure. Things won't run smoothly; the exercises aren't carefully graded; the material is poorly organized; and they just don't like it. Before a teacher passes judgment upon a general mathematics text she must review her aim and ideal in teaching and look carefully to her measures of value; subordinate some values and exalt others. The late texts in algebra and geometry are nearly perfect for their purpose. There have been so many authors and so many revised editions that the finished product is good. Texts in reorganized courses must go through the same careful development. The fact that there are imperfections in them now should not reflect against the advisability of the general mathematics course.

What the course needs now is a more critical attitude on the part of administrators and teachers. It needs thoughtful criticism and not passive adoption or rejection. High school mathematics teachers, on the whole, are well prepared to deal with the problems of organizing a first year course for themselves. Present texts and the reports of committees on the subject will give the foundation and show the proper bounds for individual experimentation. Many teachers must add good ideas to the common fund, before excellent texts can be produced. A distinguished professor of educational philosophy is credited with saying that he wished that every high school text-book might be built to last just one year, then go to pieces like the one-horse shay. If that were the case, perhaps we teachers would give up being such textbook addicts, would profit better by our mistakes and develop better judgment and greater initiative.