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2. Whitehead, A. N. Introduction to Mathematics. Home University

Library. Henry Holt & Co. $0.50. A remarkably clear exposition of the salient features of elementary

mathematics. 3. Young, J. W. Fundamental Concepts of Algebra and Geometry. The

Macmillan Co. $1.60. A treatment of the logical foundations of algebra and geometry from

the modern point of view, presuming no large knowledge of these

subjects. 4. Young, J. W. A. et al. Monographs on Modern Mathematics. Long

mans & Co. $3.00. A more scientific and detailed account of such topics as the foundations

of geometry, modern pure geometry, non-euclidean geometry, the fundamental propositions of algebra, the algebraic equation, the function concept and the fundamental notions of the calculus, the theory of numbers, constructions with ruler and compasses, the history and

transcendence of . 5. Poincaré, H. Science and Hypothesis. Science Press. $1.50; Value

of Science. Teachers College. $1.50; Science and Method. Three collections of acute and stimulating philosophical studies by the

foremost mathematician of his time: Soon to be published in one

volume by the Science Press. 6. Workman, W. P. Memoranda Mathematica. Oxford University

Press. $1.75. A collection of facts, formulae and results from arithmetic, mensuration,

algebra, theory of equations, determinants, plane and spherical trigonometry, geometry, geometrical conics, analytical geometry, differential calculus, elementary statics, dynamics and hydrostatics, with frequent summaries of proofs and notes. Invaluable to the high school teacher and the college student.

II. TEACHING OF MATHEMATICS.

1. Young, J. W. A. Teaching of Mathematics in the Elementary and the

Secondary School. Longmans & Co. $1.50. 2. Smith, D. E. Teaching of Elementary Mathematics. The Macmillan

Co. $1.00. 3. Schultze, A. Teaching of Mathematics in Secondary Schools. The

Macmillan Co. $1.25. The last three books are of unequal merit. Young's book should have

the preference.

4. Smith, D. E. Teaching of Geometry. Ginn & Co. $1,25.

Written on rather conservative lines. 5. Perry, J. Teaching of Mathematics. The Macmillan Co. $0.75. An illuminating report of a discussion on the teaching of mathematics

at the meeting of the British Association in 1901, showing serious defects in English methods.

6. Young, J. W. A. Teaching of Mathematics in Prussia. Longmans &

Co. $0.80. A little out of date, perhaps, but still valuable for the sake of com

parison.

III. ARITHMETIC.

1. Lodge, Sir Oliver. Easy Mathematics. Chiefly Arithmetic. The Mac

millan Co. $1.10. A fresh and breezy treatment of topics that ought to be easier to learn,

from arithmetic to the calculus. A book for every teacher's library.

2. Tannery, J. Lecons d'Arithmétique, théorique et practique. Colin,

Franc 5.

A fine specimen of the French theoretic presentation of arithmetic.

IV. ALGEBRA.

1. Chrystal, G. Treatise on Algebra, in two volumes. The Macmillan Co.

$8.50. Unquestionably the best English work on the subject. 2. Burnside, W. S. and Panton, A. W. Theory of Equations, in two vol

umes. Longmans & Co. $6.00. An excellent work, going into a thorough discussion of the theory of

equations, determinants, etc.

V. GEOMETRY.

1. Heath, T. L. The Thirteen Books of Euclid's Elements, in three vol

umes. Cambridge University Press. $13.50. A marvelously well-written and comprehensive book. The English

classic on the subject. It ought to be in the hands of every teacher

of geometry. 2. Taylor, H. M. The Elements of Euclid. Cambridge University Press.

$1.25. One of the best of recent English Euclids.

3. Halsted, G. B. Rational Geometry. John Wiley & Sons. $1.50. A A book written along the lines of Hilbert's Foundations of Geometry.

A good exposition of recent methods. 4. Rouché, E. and de Comberousse, C. Traité de Géométrie Gauthier

Villars. Francs 17.
Probably the most complete work in any language. The book from

which American authors have borrowed most freely. 5. Klein, F. Famous Problems of Elementary Geometry. Ginn & Co.

$0.50. A very interesting work, showing how the three famous problems of

antiquity are discussed in the light of modern investigations. 6. Adler, A. Theorie der Geometrischen Konstruktionen. Goescheneche

Verlagshandlung Marks 9. In this book the possibilities of constructions by the ruler alone, the

compasses alone, the ruler with parallel edges, the ruler and a fixed

circle, etc., are carefully examined. 7. Manning, H. P. The Fourth Dimension Simply Explained. Munn and 7

Co. $1.50. 8. Manning, H. P. Non-Euclidean Geometry. Ginn & Co. $0.75. These two books, of elementary type, may serve to interest teachers

who wish to know a little regarding the subjects discussed. 9. Vinbert, H. Les Anaglyphes Géométriques. Librairie Vinbert. Francs

1.50. A handsomely illustrated book showing how a marvelous stereoscopic

effect may be secured by looking with the left eye through a red medium, with the right eye through a green, at a figure properly drawn in red and green.

VI. HISTORY OF MATHEMATICS.

1. Ball, W. W. R. Short History of Mathematics. The Macmillan Co.

$3.25. 2. Cajori, F. History of Elementary Mathematics. The Macmillan Co.

$1.50. The former of these books, though not free from errors, is perhaps the

more readable. A high school library should have both.

VII. RECREATIONS.

I.

Ball, W. W. R. Mathematical Recreations and Essays. The Macmillan

Co. $2.75
A book sure to interest both teacher and pupil.

VIII. PERIODICALS.

I.

School Science and Mathematics. Chicago. $2.00.
A journal for science and mathematics teachers in secondary schools

which well fulfils its purpose. 2. The American Mathematical Monthly. Chicago. $2.00. A journal for teachers of mathematics in the collegiate and advanced

secondary fields. Under new auspices, with a large editorial corps, an excellent periodical is to be expected.

FIRST LESSONS IN GEOMETRY.

MISS MARION S. GERLS, DETROIT CENTRAL.

The little I shall be able to tell my hearers must, of necessity, be limited to those experiences which have come in my way, while traveling along the path of the allotted subject.

I am going to ask indulgence for a moment or two while I make a passing reference to the subject of teaching, generally. The argument is made over and over again that the teacher, to be successful in his chosen field of employment, must love the work. If this is true of the teacher generally it is especially true of the teacher of Geometry. To inspire and interest a class in this study I contend that one must first have undoubted faith and confidence in one's own ability and qualification. Without such confidence, or assurance, if you would so term it, the teacher's success with a class in geometry, as it appears to me, will not win extravagant congratulation.

During the first few days with a class of beginners in Geometry, it has seemed best, in my work at least, that very little attention should be given to a consideration of the real subject, but that the class should be led step by step to a feeling of familiarity with the environments, of acquaintance with one another, and to at least some degree of confidence in their teacher. This rule may apply in a measure to any kind of class work but it has appealed to me as exceptionally adapted to a class of beginners in geometry.

There are three important factors which have appeared to me to enter largely into the question of successfully teaching mathematics. I shall attempt to define these as, first, the attitude of the teacher toward this particular study, second, the view with which the pupil regards the subject, and third, the light in which the teacher and pupil see each other. It is my aim to try to weigh up the relative importance of these factors, and through a few days work in the beginning, with now and again some familiar talks along interesting lines, lead my class into an attitude of mind through which these conditions will readily adjust themselves.

The first lesson assigned to a class of beginners in geometry will, naturally relate to definitions of the geometrical concepts. In this lesson I find an admirable opportunity for awakening in each pupil a strong desire for independent and concentrated thought. It always appears to me one of the most efficient helps when the pupil recognizes and appreciates the importance and the need of thinking. At times I find my own ingenuity pretty well taxed in an effort to discover some original process which will force upon a certain pupil in the class, a desire to concentrate his mind upon the lesson and on the particular question we are considering. And I find it especially helpful to create in the mind of each a strong desire to be in the class on the following day.

While I assume, as a matter of course, that each pupil coming to the class knows the lesson perfectly I do not hold the class to a rigid or formal recitation, but instead, I begin with a sort of informal talk about these geometrical concepts, turning for a few moments to some familiar topic such, for example, as “Who is the author of this book we are using ?” and “Why should this author consider himself qualified for writing a book for the use of high school pupils ?" This refers them to the title page. Then I may turn the conversation to a few words along the line of copy-righting, and soon branch into a general inquiry of the class to learn how many there are able to tell what is meant by giving the definition of a thing. I call on one to define some familiar object, or to give a description of some piece of furniture in the room and thus gradually bring about the need of a definition for a solid. Our author's first statement is “A material solid occupies a limited portion of space.” This conversation will very likely bring to light the fact that a decided difference of opinion exists concerning the real meaning of "material" and that different members of the class will have dissimilar or discordant ideas and thoughts as to the meaning of "solids," "space,' and other relevant and important terms. These definitions and opinions create much general interest and awaken a lively attitude of the class toward the lesson.

When I ask a pupil if he imagines that any matter is positively solid and give the assurance that the class will soon come to an understanding of the reason why it is impossible for matter to be solid, I notice that a look of incredulity creeps over the faces of the class, and then, when I introduce the idea of vibratory action, I can see the faces light up with a glow of intense curiosity. Following this up I introduce an easy or conversational discussion upon the idea of dimensions, asking for definitions, and coming gradually to the story of the man of lineland, and of the man of flatland, thus leading up to a consideration of the man of three dimensions, the conversation soon begins to excite a lively interest. The teacher who is inter

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