and cultivate a love of honor that contestants will scorn to stoop to indirection, and will generously and sincerely applaud the evidence of the possession of desired power by another. Few would advocate or approve anything else than fair play when they were not under the pressure of strong excitement, warping their judgment as the lens is made to distort an image when subjected to pressure. The idea must be fixed in the minds of pupils that defeat is not disgrace, tho victory be glorious, and that no honor lies in success unless achieved by merit and as the result of courage and determined effort. In the individual school it is not difficult to secure a right spirit of generous rivalry. When more than one school is involved, concerted action is required, and the conscious cultivation of the true sportsman's spirit. In school organizations and interscholastic leagues, faculty, pupils, and alumni should act in conjunction, if the best results are to be secured. The desire of the school authorities must be to contribute maturity of judgment and knowledge of affairs, which pupils may not possess on account of inexperience. Pupils will not be found intractable, and where sympathy is evident will be glad to accept suggestion. They must be made to feel that, while the management of athletics is laid largely upon them, the very existence of school organizations is dependent upon the favor and approval of the principal, who is responsible for the successful conduct of the whole, and deeply interested in the vigor and growth of every part.

In my own school, so important have I deemed the conduct of athletics that I have given it in charge of the assistant principal, who devotes to it whatever time and attention may be required. While we have not accomplished all that we desire, so much has been done that I may confidently affirm that with forethought and attention athletics may be made to yield abundantly, and physical, mental, and moral development result.

DISCUSSION

The discussion which followed was full of life and interest, and was carried on freely for an hour and a half, both ladies and gentlemen participating.

It was the general opinion of those present, who represented more than twenty states and territories, that the real boy and girl can be reached most effectively thru properly regulated athletics.

Many helpful suggestions were made by men from various localities. The Iowa plan of regulating school athletics was explained, and also the plans of Illinois, Wisconsin, Minnesota, and St. Louis, Mo.

The possible usefulness of a federation of state organizations in this connection as well as others was suggested.

The salutary influence of well-regulated school athletics upon college athletics and upon those likely to engage in them was clearly shown.

The theme of the paper proved so fruitful of discussion that there was not time to consider the other questions selected by those present.

MATHEMATICAL CONFERENCE

LEADER -CHARLES W. NEWHALL, SHATTUCK SCHOOL, FARIBAULT, MINN. TOPIC: CORRELATION OF MATHEMATICAL STUDIES IN SECONDARY SCHOOLS

THE LEADER: Mathematical instruction in our educational system is divided into the three large classes: elementary instruction, secondary instruction, and higher instruction.

In attempting to secure correlation of the various branches of mathematics into one homogeneous whole, we should first seek to find this unity extending thru all these

divisions of educational activity. By a careful adjustment of the work of these three divisions much of the waste in mathematical instruction may be avoided.

If the secondary schools and the grade schools can agree that certain parts of arithmetic be omitted from the courses of the grade schools, and elementary algebra and geometry be introduced in their places, we will find that the pressure in the secondary schools will be much relieved and the student will enter the high school much better equipped for his work. Then if, on the other hand, the schools and the colleges can agree as to just what part of the work in mathematics shall be done before entering college, our first great work of harmony and unity is effected.

Fortunately this bids fair to be soon accomplished. The Committee of Ten of this Association, and more recently the Committee on College Entrance Requirements, have made specific recommendations looking to exactly this end. The Committee on College Entrance Requirements has made a long step forward in defining the units of work and the character of the instruction that shall be required in mathematics as a preparation for college. The Examining Board of the Middle States and Maryland has tended to crystallize sentiment in the matter by the prompt adoption of their recommendations.

Whether or not all colleges accept the plan of this board, and whether the entrance to college may better be by certificate than by examination, the fact still remains that the work of this committee has practically made definite and uniform the amount of mathematical training a candidate for college must acquire. So we may consider that the waste and friction at the point of division between the secondary and higher schools are in a way to be satisfactorily eliminated.

On the other hand, educators are equally unanimous as to the best way of securing harmony between the work in mathematics in the grades and in the secondary schools. We all agree, I am sure, as an academic proposition, that geometry and algebra in elementary form should be begun in the grade schools. When the fortunate time comes when we shall have six years in our high-school course, we can then arrange these matters to suit ourselves; but at the present time we are presuming to have the same right to require from the grade schools certain preparation for our work that the colleges exact from us in preparation for their work.

I believe the grade-school teacher and the superintendent feel the necessity and the advantage of these mathematical studies in the grades as much as the secondary teacher does, and the reason the idea has not been more generally adopted is on account of the practical difficulties involved. In the Chicago public schools the experiment of teaching algebra in the grade schools was tried and abandoned, partly, I believe, on account of the inexperience of the grade teachers. In other places it has been found impossible to introduce any new studies into the already crowded curriculum of the grammar schools, or to require any new work of teachers already teaching too great a variety of subjects. In some places there exists a prejudice against displacing such a practical study as that of arithmetic to make room for the study of geometry and algebra. But these prejudices and practical difficulties have been overcome in many places, and the wisdom of introducing the study of elementary geometry and algebra in place of certain special subjects in arithmetic has been conclusively demonstrated. It will not be long before all schools will find it necessary to readjust their curricula so as to conform to these recommendations.

When this time does come, the teacher of mathematics will be fortunate beyond most of his fellows, except perhaps the teacher of English. He will then have a course in mathematics beginning in the kindergarten, extending thru the primary schools, the secondary school, and at least the first two years of college.

If the students from the kindergarten to the college are to be turned over to us a certain number of hours each week of every year for mathematical instruction, why can we not map out a broad and comprehensive scheme for the study of mathematics, which shall call for a gradual development and a constantly broadening view of the whole field of

mathematics and which shall not be limited by such artificial boundaries as are at present set to the study of algebra, geometry, trigonometry, etc.?

In many schools algebra is studied, say, in the first year of the high school, then discontinued while plane geometry is taken up; this is completed, and then perhaps higher algebra is studied, or solid geometry, and, where the school offers it, the student continues with trigonometry and the use of the logarithm table. Is there any necessity for these artificial breaks? Is it best to complete one subject and then lay it aside on a shelf to rust while we take up another, even if we may find time later to take down the first to brighten it up a bit, to pass the inspection for college?

The Committee on College Entrance Requirements is very clear in recommending that the study of geometry and algebra should be carried on in the same year, perhaps geometry two days a week and algebra three, or geometry the first half-year and algebra the second, but at any rate the two studies in the same year. But can we not do better than this? Are not the subjects intimately enough related so that we can teach them both the same day and every day? We can thus carry the study of geometry and algebra along together thru several years, introducing also the simple ideas of trigonometry early and developing that subject alongside of the others. We may even introduce a few elementary ideas from the analytic geometry at places where they will elucidate other

matters.

Such an arrangement of subjects would call for no more time than we at present devote to them all separately. We could teach the same matter in four years, or six, but the arrangement would be different; in fact, there would be a great saving of time and of effort in conquering a difficulty once instead of two or three times in slightly different aspects as it presents itself in the different studies.

We could make the study of a geometric figure throw light on the solution of an algebraic equation, we will say, or, conversely, we could use the tools of algebra to help us out of some geometrical difficulty. Some of us do that now. Yes, but the scheme is capable of a much wider development. It has been found successful in Germany. There the various lines of mathematical study are carried along side by side and interwoven into Each subject is studied thru several years with constant reference to its bearing upon and connection with each of the others. The result attained by their system seems to be better than the result attained by ours, and it is accomplished in a shorter time.

one.

This correlation of the mathematical studies begins very easily in the elementary schools. In the kindergarten the child begins to acquire his ideas of number and of form at the same time. Thru the elementary school a skillful teacher can combine the study of drawing, arithmetic, and mensuration so as to leave a well-rounded and definite impression in the child's mind as a result of this training. The equation can be introduced very early. If the problem to be solved by the use of the equation is a numerical problem the introduction of x will cause no difficulty. The child will soon see that the x is merely an abbreviated form of writing "the tail of the fish," or whatever may be the subject of the problem.

After algebra has been begun arithmetic should not be discontinued. The study of the operations with fractions in algebra can be illustrated by similar operations with arithmetical fractions. An algebraic problem can be verified and made concrete by substituting numerical values. Or the general nature of an algebraic formula may be shown, by applying it to a great many special arithmetical cases. Or we may study the answers to a general problem in which the known numbers are represented by letters, and discover how each one of them enters into the result. Then we may solve the same problem where the given quantities are numbers and find that the various known elements of the problem are all merged into one number and lose their identity. By the use of a little skill, the teacher can show that a literal problem is much more interesting and instructive than a special numerical case of the same problem, inasmuch as we are able to trace every given quantity thru the work of the solution and find how it affects the result.

Students are not slow to see that algebra is merely a generalized arithmetic; that the operations are the same, and that the forms of the results are different only because in algebraic solutions we can often only indicate our operations, and cannot actually perform them so as to merge several quantities into one, as with arithmetical numbers.

I am sure every teacher of geometry has felt hampered at times because the student does not understand a simple algebraic manipulation that becomes necessary in connection with some geometrical proof. He may have forgotten the algebraic principles involved, or he may not feel quite sure of his ground in applying his algebra to such an unfamiliar problem. If the study of algebra and geometry were carried on together, neither of these difficulties would occur. Many problems in computation in geometry are quite easily worked out by the use of equations, or certain proofs involve a little algebraic manipulation. The student should feel perfectly sure of his algebra, so that the geometrical truth should not be obscured by any haziness about the algebra involved.

Geometry and algebra are entirely different, of course, in their subject-matter. The one deals with number as expressed by various symbols and the other deals with form. But they touch at many points.

For example: Take the formula (a+b)2 = a2+2ab+b2 and interpret it geometrically into "The square of the sum of two lines is equivalent to the sum of their squares plus twice their rectangle." Similarly we can show geometrically the truth of the algebraic formula (a — b)2 = a2 — 2ab+b2 and (a — b) (a + b) = a2 — b2. If the same results are arrived at in two ways, by an algebraic and a geometric consideration, how much light one method will throw upon the other!

Again, would it not be a good plan to introduce the notion of a graph in connection, with the solution of quadratic equations ?

If the student can see that the straight line cuts the parabola in two places, and then learns that these two places give the values of x and y found by solving the equations of the two curves considered as a pair of simultaneous equations, I am sure he will have a clearer idea of the fact that altho either of our given equations, alone, may be satisfied by an infinite number of values for x and y, there are only two pairs of values for x and y that will satisfy both equations at the same time. The meaning of simultaneous equations becomes apparent. And again, I am sure he will readily see by a study of graphs that a system of two equations, one of the second degree and one of the first degree, will give two values for x and two for y, while a pair of quadratic equations will in general give rise to four values for each. And if he plots various curves at the same time that he is solving for the roots of their equations it cannot help but be instructive to examine the cases where a given equation has negative roots, equal roots, or imaginary roots. To illustrate these algebraic facts by the use of graphs would require only very elementary ideas of analytical geometry.

When we are considering similar triangles and proportional lines, why can we not learn the names of the trigonometric functions? The pupil learns that in all similar right triangles the ratio of one leg to the hypotenuse is always the same. Why not define this ratio at once as the sine of the angle A, and similarly define the other trigonometric functions? There can be no better place to introduce these ideas. And again, when the student is learning, as in Book III of most of the geometries, the relations that exist between certain lines of certain triangles, why not initiate him at once into the relations which exist between the sides and angles of any triangle? A bright student will almost always raise the question whether the sides of a triangle are not connected by some relation with the angles of the triangle. The introduction of these fundamental notions would not call for much trigonometry at this point.

Why not follow up the chapter on exponents in algebra by a study of the simpler properties of logarithms? A logarithm is but an exponent, and the mystery and difficulty surrounding the operations with logarithms would disappear if the student could be made to see this fact clearly and to recognize that the familiar laws for exponents govern

the work with logarithms. The student will grasp eagerly the notion of the practical value of a logarithm table. A few calculations with logarithms would give practice in arithmetic and an excellent review of decimals, and perhaps accustom the student to the use of a table of logarithms in other subjects besides that of trigonometry. As taught at present, there is almost always in the student's mind a confused idea that logarithms have some intimate relation with trigonometry. They are never quite sure of the difference between the tables of natural functions and the tables of logarithmic functions.

I have suggested but a few places in which the studies of arithmetic, algebra, geometry, and trigonometry may be brought together. Any teacher can think of many other ways in which the various subjects can be made to help one another.

I have said nothing about an attempt to correlate the sciences with the mathematical studies. The study of physics and mathematics, for example, are so inter-related and interdependent that it is hardly possible to separate them. Geometrical and arithmetical progressions are illustrated from physics. Physics contributes many interesting problems to geometry and algebra, and geometry and algebra are often called in to help develop the laws of physics.

Of course a broad application of the principle of correlation would call for a correlation of physics, astronomy, and other of the sciences with mathematical studies, which broad acceptance of the term would be an excellent thing, and in every way to be encouraged. It would be well, for example, if mathematics and physics could be taught by the same teacher, so he could elaborate a complete correlation between them.

I have not attempted to consider here the broader view of the correlation of mathematical studies with other subjects, but rather the more definite and limited question of the correlation of the various branches of mathematical study with one another. The possibility of merging the subjects of arithmetic, algebra, geometry, and trigonometry into the one broad subject of mathematics, and teaching it as such thru all the years of the elementary and secondary schools, is the question I want to propose for your consideration and discussion.

Of course the attempt to follow out this plan in its full and ideal extent would involve the working out of a new system of instruction, a new allotment of subjects to the various years, new text-books perhaps. If the plan is a wise one, a good teacher can meet these difficulties. If the idea is too visionary for a full materialization, may we not gain something by a partial application of the principle?

I am putting the question for your consideration, then, we will say, in two parts: (1) Is the plan as suggested here, in its full and ideal development, a sound one, and (2) To what extent is it practical, at this time, to work it out?