in x and y is solvable. Furthermore, many pupils fail to see the necessity of combining each r with its appropriate y-they fail to recognize that it takes a pair of numbers to effect a solution. Once the pupil realizes, however, that the process of solving a group of simultaneous equations consists, essentially, in successively replacing one group by an equivalent group of simpler equations, he is likely to escape the danger of coming out, at the end of his calculation, with a list of 's and y's before which he stands helpless until he has tested them all. The use of graphical methods is almost indispensable, here, in ordering the pupil's commonly confused notions.

The teaching of mathematical induction seems peculiarly difficult. Even after the pupil has been convinced that no general law can be established by special cases, he persists in going astray. He assumes the law to hold for n cases, and shows that it then holds for n+1 cases. There he stops, thinking the proof complete. He fails to identify n with a known case, so as to eliminate the assumption.

It is perhaps unfortunate that the abstract notions of limit and infinity should present themselves to high school pupils; but these notions are indispensable to an adequate discussion of the trigonometric functions, or of the quadratic equation. The definition of limit in the text-books is usually adequate for elementary needs, but the illustrations are frequently misleading, in that the pupil is persuaded that the function is always different in value from the limit, even that it is always smaller than the limit; and these false notions are sometimes supported by the series +++.., the sum of n terms of which is, to be sure, always different from, and smaller than, the limit of that sum, viz., unity. In the progression 3-+-+.., however, the sum of ʼn terms is alternately larger and smaller than the limit. Neither teachers nor text-books always distinguish between the limit of f(r) when r approaches a, and the actual value of f(x) when equals a. If, for example, ƒ(x) f(x)=4, but f(2) has no definite meaning. On the other hand,

8

22-4
x-2

> then

lim

x=2

im f(x)=3 and ƒ(1)=3 also. If however, one puts f(x)

1

= sin and allows to approach zero, f(x) assumes every real value between +1 and -1, over and over again, without ap

lim

proaching any fixed limit. Thus neither f(x) nor f(o)

x=0

exists. This well-known example illustrates the possibility, that the usual text-book definition of limit may be satisfied, under circumstances where, nevertheless, a limit does not exist. The difference sina, where a is any fixed number between +1 and I, will become smaller than any preassigned positive number e, as r approaches zero, but it does not remain smaller than e.

The notions that the high school pupil gets from class, discussions of infinity are not infrequently grotesque. Infinity becomes for him a constant, so large that it must be mentioned with awe; or a mysterious something to haggle over, and to furnish paradoxes. This seems unnecessary. The situation that arises, wherever this concept presents itself, is the following: Under certain circumstances, a variable number, r, increases in such a way that it can exceed, in absolute value, any preassigned constant. This fact is expressed conventionally, by saying that

π

becomes infinite, under the assigned circumstances. The example most familiar to the pupil is the trigonometric ratio, tan Ꮎ. If the angle → becomes the tangent does not assume a definite value-on the contrary it ceases to be a definite number, the defining ratio loses its meaning. But for all values of e±

2,

π

2

π

2

the

tan has a definite meaning, and as e approaches variable, tan e, may be made to exceed, in absolute value, any constant previously agreed upon. Current text-books are partly to blame for the pupil's hazy notions, when the concept infinity is involved; teachers are to blame for allowing errors of the text to pass unchallenged. In one text, we read that r becomes infinite when it exceeds "any conceivable number"-a manifest contradiction. From another text, we learn that infinity is the number in spite of the author's earlier statement that division by zero is meaningless.

1

Teachers fail to make it clear that, under the above definition, infinity must be a variable. The necessity for the "preassigned” number of this definition being a constant, is likewise overlooked. becomes infinite; but it cannot be made to

E. g. when x=0,

1

exceed Yet x2

1

1

x

x2 would serve perfectly as the comparison number of some texts and of many teachers-it is just as "con

Ambitious teachers sometimes include, in the school curriculum, one of the courses in advanced algebra (numbered 12a3, 12aʻ in the matriculation list of the University of California), although the topics concerned are too difficult for the majority of pupils in the average school. A four years' course in mathematics reads well in the school announcement, and the well-trained teacher finds the exposition of the theory of equations, convergency of series, continued fractions, etc., a pleasurable occupation. But in some cases the intellectual health of the school is sacrificed by encouraging, or permitting, pupils to take these subjects, who are unable to master the difficulties involved. If classes are to be formed in 12a" or 12a', they should be of selected pupils, of the senior year; and unless the courses can be given in a thoroughgoing way, they should not be given at all. The writer's personal conviction is, that the University of California should withdraw 12a' and 12a' from its published list of matriculation options, and so discourage their being attempted, except in response to a normal demand.

In conclusion, the writer would like to offer a few suggestions to inexperienced teachers, as to the conduct of recitations. Every teacher should make thoughtful preparation for each class exercise. However well he may know his subject, he owes it to the class, that he should organize his material, and so plan the recitation, that appropriate stress may be laid, and the greatest profit accrue to the greatest number. It is a common mistake for the novice to look upon a recitation primarily as a test of the individual who is reciting. This is not teaching. The reciting pupil is rather a foil for the instruction of others, himself learning, at the same time, oral exposition and all that goes with it. This means that only those propositions should come up for discussion that are most significant, or that involve difficulties which must be cleared away. It means that the reciting pupil should have the attention of the class-not merely that he should talk to the teacher while his neighbors are getting ready for their turns. The recitation period is short. It belongs to the class as a whole, and the teacher should see to it that every moment is profiting the largest possible number. No part of it should be devoted exclusively to individual instruction. If a pupil needs something that nobody else wants, he might get it from the teacher at another time or from some fellow pupil. Too much time is consumed, during recitation hours, in writing upon the

blackboard. It is characteristic of a large number of teachers, to start a class exercise by sending a troop of pupils to the board, where from a third to a half of the period is spent in writing a demonstration or the solution of a problem. Perhaps those in their seats are given a problem to keep them busy. This is virtually wasting time for all in the class. When those at the board have finished, they are required to read aloud what they have written a deadening process, because it lacks spirit and spontaneity. How much more interesting and profitable it would be, if the pupil, before reciting, should put upon the board merely a figure, or the skeleton of a demonstration; or, still better, why not have a pupil start with a clean board, and put down figures and equations as necessity for them develops? This would be the ideal of a comprehending and instructive exposition. It would serve toward overcoming what we all recognize as the most regrettable lack in our pupils-the inability to express themselves in pertinent, intelligent English.

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GRAPHICAL ILLUSTRATION OF CONVERGENCE OF SERIES.*

S. EPSTEEN AND F. W. DOOLITTLE,

University of Colorado.

Experience shows that beginners find the chapter on Convergency and Divergency of Series one of the most difficult in the entire subject of elementary algebra.

The following graphical presentation has proved very helpful in the Freshman Engineering Algebra of the University of ColoThe treatment is not, and could not be made, rigorous; it is intended merely as illustrative, in order that the students may have before them the concepts and theorems in pictorial

rado.

as well as in algebraic form.

Definitions and examples are given in order that this paper may be a self-sufficient chapter for teachers who deem a greater amount of rigor not desirable when covering the subject for the first time. Our own method consists in having this graphic presentation supplement rather than supplant the text-book.

Infinite Series. If u1, u2, uz, un denote any given never ending sequence of numbers, the expression untu2+μз+ tunt . . . . . is called an infinite series. For u1+u2+..... we may write Σ, read "sum of un to infinity."

The series is called real when all its terms u, u, U3,.

are real, positive when all its terms are positive.

A series is often given by means of a formula for its th

term, un.

Thus if un Vn, the series is

=

n+1

EXAMPLES.

Write down five terms of the series whose nth terms are,

*Read before the Colorado Mathematical Society, Denver, Colorado, May 11, 1907.