Publications of the University of Texas Publications Committee: FREDERIC DUNCALF C. T. GRAY KILLIS CAMPBELL E. J. MATHEWS D. B. CASTEEL C. E. ROWE F. W. GRAFF A. E. TROMBLY The University publishes bulletins six times a month, so numbered that the first two digits of the number show the year of issue, the last two the position in the yearly series. (For example, No. 1701 is the first bulletin of the year 1917.) These comprise the official publications of the University, publications on humanistic and scientific sub­jects, bulletins prepared by the Bureau of Extension, by the Bureau of Government Research, and by the Bureau of Eco­nomic Geology and Technology, and other bulletins of gen­eral educational interest. With the exception of special num­bers, any bulletin will be sent to a citizen of Texas free on request. All communications about University publications should be addressed to University Publications, University ef Texas, Austin. 1572-7 544-4-18-21-1600 University of Texas Bulletin No. 2127:-May 10, 1921 The Texas Mathematics Teachers' Bulletin Volume VI, No. 3 l'UBLISHED BY THE UNIVERSITY SIX TIMES A MONTH, AND ENTERED AS SECOND-CLASS MATTER AT THB POSTOFFICE AT AUSTIN, TEXAS, UNDER THE ACT OF AUGUST 24, 1912 The benefits of education and of useful knowledge, generally diffused through a community, are essential to the preservation of • free govern­ment. Sam Houston Cultivated mind is the guardian genius of democracy. . . . It is the only dictator that freemen acknowl­edge •nd the only security that free­men desire. Mirabeau B. Lamar University of Texas Bulletin No. 2127: May 10, 1921 The Texas Mathematics Teachers' Bulletin Volume VI, No. 3 Edited by P. M. BATCHELDER Instructor in Pure Mathematics, and A. E. COOPER Instructor i!l Applied Mathematics This Bulletin is open to ttle teachers of mathematics in Texas for the expression of their views. The editors assume no responsibility for statement!' of facts or opinions in articles not written by them. PUBLISHED BY THE UNIVERSITY SIX TIMES A MONTH AND ENTERED AS SECOND-CLASS MATTER AT THE POSTOFFICE AT AUSTIN, TEXAS. UNDER THE ACT OF AUGUST 24, 1912 CONTENTS Acknowledgment;......................... ..The Editors........ 5 Courses in Mathematics in the Summer School. ............. . ............ . .......H. J. Ettlinger. . . . . 6 The Original Exercise in Geometry...... .....J. G. Dunlap ....... 9 Some of the Outside Work Done by Students in the Marshall High School . .. . .... ... .... Paula Henry .. ... ..15 Logarithms ................................ P. H. Underwood ...19 A Few Suggestions I Have Found Useful in Teaching Plane Geometry................. Mrs. Lee Brinton ...27 "Stratified or Correlated Mathematics-Which?" . ................................Dora Willingham ...32 Less Memory Work and More Thought in Geometry................................Helen Carr......... 36 .. MATHEMATICS FACULTY OF THE UNIVERSITY OF TEXAS P. M. Batchelder H. Y. Benedict J. W. Calhoun C. M. Cleveland A. E. Cooper Mary Decherd E. L. Dodd H. J. Ettlinger Helma L. Holmes Goldie P. Horton Jessie M. Jacobs J. N. Michie R. L. Moore Anna M. Mullikin M. B. Porter C. D. Rice It is with great pleasure that the Bulletin welcomes six new contributors to its pages. Mr. Dunlap is the prin­cipal of the Cleburne High School; Miss Paula Henry is a teacher of mathematics in the Marshall High School; Mr. Underwood is the mathematics teacher in the Ball High School, Galveston ; Mrs. Lee Brinton is teaching mathematics in the South End Junior High School of Houston; Miss Dora Willingham is of the Brownsville High School; and Miss Helen Carr is principal of the Orange High School. Their articles will be of real practical value to many other teachers. Consequently, the Bulletin considers itself fortunate in being the medium through which they may express their ideas on teaching methods; and it awaits with interest any other such material they may care to con­tribute. COURSES IN MATHEMATICS IN THE SUMMER SCHOOL The courses listed below are offered in the Summer Ses­sion of 1921 of the University of Texas. Courses counting towards a teacher's certificate may be taken. All courses count toward the B. A. degree and some may be counted to­ward the M. A. and Ph. D. In addition to the class work, Professor Earl Raymond Hedrick, head of the Department of Mathematics in the University of Missouri, has been se­cured as a special lecturer. He will deliver a series of lec­tures over a period of one week on "The Teaching of Sec­ondary School Mathematics." These lectures will be given some time during the first term of the summer session. Professor Hedrick is the mathematical editor of the Mac­millan Company for high school as well as college text books. He has also taken an active part in the work of the National Committee on Mathematical Requirements and is recognized as an authority in his field. Every high school teacher in Texas should take advantage of hear­ing this course of lectures. FIRST TERM Students who present trigonometry for entrance may not register for Mathematics la; those who present solid geom­etry for entrance may not register for Mathematics lx. Mathematics la, or entrance trigonometry, is prerequisite to le; and le is prerequisite to ld. la. PLANE TRIGONOMETRY.-This course will cover the subject of trigonometric functions of angles, identities, so­lution of triangles, inverse functions, circular measure, and logarithms. The arithmetic side of the subject will be em­phasized. There will be much problem solving. Three sec­tions. Associate Professor Rice; Instructors Horton and Mullikin. lb. ALGEBRA.-This course will assume a knowledge of the amount of algebra usually covered by a good high school, but will review some of the topics of the high school Mathematics Teachers' Bulletin course. Especial stress will be laid on quadratic expres­sions and equations, the graph, logarithms, progressions, determinants, and the binomial theorem. Three sections. Adjunct Professor Michie; Instructors Batchelder and Mullikin. le. INTRODUCTION TO ANALYTIC GEOMETRY.-This course will be devoted to a brief consideration of Cartesian co­ordinates, plotting curves from their equations, the analytic geometry of the straight line and of the circle. Prerequi­site: trigonometry. Associate Professor Rice. ld. ANALYTIC GEOMETRY.-This course, to which le is a prerequisite, will make a very brief review of the straight line and the circle, and will be mainly devoted to a consid­eration of the analytic geometry of the parabola, the ellipse and the hyperbola. Professor Porter. lx. SOLID GEOMETRY.-Adjunct Professor Michie. 3a. A. CALCULUS.-In this course the fundamental prin­ciples of calculus will be considered. Prerequisite: three­thirds of Pure Mathematics 1 or of Applied Mathematics 1 (exclusive of solid geometry) or Pure Mathematic!'i 215. Professor Benedict. 3b. or 3c. A. CALCULUS.-A second term in calculus. Prerequisite: Pure Mathematics 3a, 3b, or equivalent. Pro­fessor Benedict. 102. A. FUNDAMENTALS IN ELEMENTARY MATHEMAT­ICS.-For students interested in the philosophy or the ped­agogy of mathematics. The significance of the number­concept; some important proofs in algebra; brief survey of the foundations of geometry ; geometrical construction ; de Moivre's theorem; and trigonometry. Prerequisite: two courses in mathematics. Associate Professor Moore. 205a. i. ALGEBRA.-This course will include complex numbers, the elementary theory of equations, the solution of higher equations, symmetric functions, etc. Prerequi­site: three terms of Mathematics 1, including lb. Instruc­tor Batchelder. lOa. i. A. MODERN GEOMETRY.-An introduction to pro­jective geometry. Prerequisite: two courses in mathe­matics. Instructor Horton. University of Texas Bulletin llc. i. s. INFINITE SERIES AND PRODUCTS.-Prerequisite: Pure Mathematics 3 or its equivalent. Professor Porter. 13a, b, or c. i. s. ADVANCED MATHEMATICS.-Prerequi­ site: two courses in mathematics. Associate Professor Moore. SECOND TERM la. PLANE TRIGONOMETRY.-As in the first term. Pro­fessor Riley. lb. ALGEBRA.-As in the first term. Instructor Holmes. le. INTRODUCTION TO ANALYTIC GEOMETRY.-As in the first term. Instructor Decherd. ld. ANALYTIC GEOMETRY.-As in the first term. Instruc­tor Holmes. lh. INTRODUCTION TO STATISTICS.-Prerequisite: Pure Mathematics lb and le or equivalent. Assistant Professor Kline. 205b. i. ALGEBRA.-A continuation of Pure Mathematics 205a, including such topics as exponential and logarithmic functions, determinants, permutations and combinations. Prerequisite: Pure Mathematics 205a. Instructor Decherd. 3b. A. CALCULUS.-The work of this course is based upon a knowledge of the calculus given in Mathematics 3a, and is a continuation of that course. Professor Riley. 3c. A. CALCULUS.-Prerequisite: Pure Mathematics 3b. Assistant Professor Kline. 225a. i. DESCRIPTIVE GEOMETRY.-Prerequisite: Pure Mathematics 1. Adjunct Professor Ettlinger. 13a, b, or c. i. s. ADVANCED MATHEMATICS.-Prerequi­site: two courses in mathematics. Adjunct Professor Ett­linger. A course in advanced mathematics is listed for each term. The subject matter of the course will be determined by the students who elect to take the work. Anyone who is inter­ested in taking advanced courses should write to the Chair­man of the Department of Pure Mathematics. H. J. ETTLINGER. THE ORIGINAL EXERCISE IN GEOMETRY The "original exercise" in this paper means that group of geometric principles whose truth must be established and problems to be solved as distinguished from the theo­rems demonstrated in the text. The use of the original exercise in impressing and emphasizing geometric truths is of comparatively recent date. The experienced teacher will find it an extremely fertile field for developing accunte thinking. The earliest manuscripts in geometry were of course very primary and the necessity of some means of fastening in mind the fundamental principles not so great. As the science was gradually developed from the old Eucli­dean scroll, the field being extended, the application o( the principles then known called for and brought into use the exercise. The demonstration of a theorem must in form be essen­tially deductive or inductive-synthetic or analytic. Each has its peculiar use, and, to some extent at least, involves the other. Analysis to discover and synthesis to demon­strate the truth or falsity of the exercise. In the !::olutio': of practical exercises and problems the ability to investi­gate and reason for one's self is the necessary prerequisite to success. This ability is not inherent in the pupil, but must be acquired by long and earnest hours of application to study. Happy should that teacher be, if he can on the one hand awake interest and on the other hold the pupil to the task of solving the tedious original. And at this point I be­lieve the pupil gets his most lasting benefit-a doggedness of purpose, a determination to win. It has been my observa­tion that most of the failures in geometry are due to a lack of tenacity. Nothing in the field of secondary mathematics is quite as good in developing tenacity of purpose as the mastery of the original-nothing quite so good in develop­ing the ability to concentrate the mental faculties. I am convinced that, if there is a superiority of the German over the American child in mathematical development, it is due largely to the preponderance of the exercise in the German text.. Too many pupils memorize the proof, if given in full in the text, thus relying entirely on the memory and throt­tling the reason. Nothing could be more harmful than this process and yet I realize that it is one of the very things we have to fight, and is especially prevalent among be­ginners. Nothing helps like the knowledge that one must depend entirely upon himself. The original furnishes this field of activity as in my opinion no other does. Smith says "The great value of teaching originals is in developing the power to think along correct lines of logical thought; if properly handled they make the pupil think more intensely and interestedly than any other subject fitted to pupils of the same age." The subject matter of the original appears in so many forms that its mastery involves a many-sided view of the subject. The pupil must be resourceful, and if blocked in one avenue of attack, turn to another. This brings confidence in himself and the ability to do something for himself as compared with the dependence manifested in the text. When once the pupil feels the joy of having accomplished something for himself he has, indeed, a stim­ulus of no mean account. Too many pupils feel overwhelmed with the apparently impossible task and surrender. Sisson says "Without enthusiasm no mathematics. Geometry is a human book-not divine-therefore, a very imperfect book. Geometry is the product of the human mind and not of the hand: therefore the subject concerns the intellect and is not mechanical." With this viewpoint, which is undoubt­edly correct, the aims which are several may be reduced to one all important one-the training of mind to habits of correct thinking and reasoning logically and accurately to a correct conclusion. Nothing in all the high school course is more important in developing the power to express one's thoughts concisely and elegantly than the original in geome­try. I dare say the student of English has found few agencies in the correct and direct expression of his thoughts more helpful. Three-fourths of its value is disciplinary. The mind should direct the hand. Mathematical reasoning is that process which step by step arrives at a definite conclu­sion. The original, as its name indicates, has within itself Mathematics Teachers' Bulletin the suggestions of the line of reasoning that leads to a solu­tion. At this point are brought to play those powers of observation which must suggest to the beginner the tools needed and the method of attack. To the beginner this is the most difficult point-"getting started"-"finding out what you want to do." These and kindred statements express the pupil's conception of the task before him. In this step by step process each is made up of two distinct parts-the statement of a fact that leads the mind in the proper direction toward a conclusion and the authority for such step. When the beginner has recog­nized this truth he has made at least one step in solving the exercise. One pupil who can and does master with fair accuracy the originals as the class proceeds instantly be­comes the leader. And in this the value of the original is recognized by the class. If this leadership is properly di­rected by a skilful teacher, growth and development of a friendly rivalry will assert itself and the effect on class work will be beneficial. The grouping of originals immediately after the basic theorem is a step in the right direction and is productive of splendid results. It enables the student to center-fire, as it were, and holds him on the principle until it is mastered. Many pupils fail utterly because they are uhable to translate English into geometric terms, thereby losing the meaning of the exercise. Again, many lose sight of the all important motive for studying geometry-the training of the reasoning faculties. They take for granted certain relations because it looks that way and are unable to pick up the train of thought when the statement is chal­lenged. This weakness can be corrected by persistent efforts on the part of the teacher in having the pupil attack the problem from another point. Our work in geometry is impaired and our success often disappointing because we omit the originals. The original furnishes the material for practice in the clinic and many originals with a few well digested theorems will bring better results of a permanent nature than many theorems and few applications of them. While it is true, as Loomis says, that problems do not con­stitute a necessary part of the science of geometry, form­ing no part of the chain of connected truths embodied there­in, yet because of their importance as applications of geo­metric principles, they are of the utmost practical value and should be studied in connection with the theorems upon which they depend. After all, may it not be truthfully said that a pupil's knowledge of geometry is measured by his ability to solve the origin"al? It is doubtful whether it is best to demand that the proof shall be submitted in a set form without variation. One of the chief ends to be attained is clearness of expression. However, it can not be attained at once. Would it not be better to be master of facts than a slave to the form? As the pupil advances the polished form of proof .should ap­pear. If the demand for correctness of form is urged too far, the pupil loses the unfolding of the exercise in his eagerness to be correct in form. Young says, "Not all ele­gance and. verbal accuracy that are to be attained later need be inflexibly required at first." Much of our trouble in teaching this part of geometry is caused by rushing the pupil over a mass of truths without time for digestion on his part. "Better plod at first and rush later," should be our motto. Many times we have heard the expression, "I just know it is so," and kindred expressions, given as reasons for certain steps taken in demonstration. To ac­cept such statements is not wholly bad, for it shows on its face that the pupil has within him the capacity for reason­ing. The reason will come later, if he is properly directed. Much of our labor has been lost because we have not di­rected the mind of the pupil in the proper way. Nothing, in my opinion, in the high school course calls for more patience on the part of the teacher than the direction of a slow student in the development of the original exercise. Again, no fixed rule of attack can be laid down, though some suggestions may be helpful. The first thing, of course, is to get a clear conception of the exercise. Skilful ques­tioning on the part of the teacher will give the student a start, as he calls it, and will help him discover the relation tetween the hypothesis and some theorem already learned. At thi::; point the imagination is brought into use. Unless Mathematics Teachers' Bulletin the pupil catches the spirit of geometric analysis he will not succeed in finding the proof. Analysis is the soul of inter­pretation and interpretation the key to the solution of the original exercise. Slaught says, "Many a high school pupil who can play the game of hypothesis and conclusion-and likes it-has never recognized geometry as a basic fact in the tiles on the floor, in the decorations on the walls, in the arches and windows of great buildings, in fact in all the mechanical and architectural developments of this and of every age. When once he comprehends this, when he takes his 'orig­inals' from actual conditions about him, then the 'game' assumes a new significance to him and geometry becomes a fact instead of a theory, a part of life instead of a mere school creation." Whether or not the student is able to reach the end just mentioned depends largely on the teacher -whether or not we recognize several distinct periods in the mental development of the child and the adaptation of sub­ject matter and methods of teaching. This means in geom­etry a readjustment of subject matter, a more accurate classification of basic truths and a multiplication of orig­inals for the application of these truths. In fact, the hu­manizing of the subject will do much to improve the suc­cess in teaching it-make the originals as practical as pos­sible, thus appealing to his observation and experience for material. Plato's school of geometry was for mature men. Plato and Euclid, if living, would be astonished at our methods of teaching geometry and at the personnel of the average class. Yet with all the advancement of the subject, it can be made more practical without losing prestige as a subject which develops and trains the mind in accurate thinking and logical reasoning. At this point in the pupil's develop­ ment the teacher must do more than ask questions. He• must bear in mind that the pupil in the beginning of a subject does not receive and understand as quickly as the teacher. He must remember that the pupil does not gen­ eralize, but must learn to do so; that the subject is not de­ University of Texas Bulletin . veloped, but is in the process of being developed. We as teachers are much given to "overshooting," as it were, the pupil-forgetting that he is not a man and therefore does not reason as one. After all, "the pupil must learn to do by doing" and will succeed if properly directed. J. G. DUNLAP. [Mr. Dunlap is principal of the Cleburne High School.] SOME OF THE OUTSIDE WORK DONE BY STUDENTS IN THE MARSHALL HIGH SCHOOL Mathematics seems to be the subject that is being dis­cussed so much at present, as to its practical value. The pupils in the Marshall High School, Marshall, Texas, too, seem to have "taken up" that thought. I have adopted the following plan : I try to finish each of the books I teach : algebra, geom­etry, arithmetic, and trigonometry, three weeks before the term closes. If the teacher shows enthusiasm the pupils, too, unconsciously show an unusual amount of interest. During the term I give a quiz a week, and then an ex­amination once every six weeks covering the work covered during that length of time. This helps firmly to "implant" in the pupils' minds the principles learned. Since they know that the problems I give them are original, they know they must make the fundamental habitual adjust­ment in the solution of the applied or practical problems, and not merely memorize rules. I have geometrical designs on cardboards, hung on the walls of my room at school. I also have a number of ex­cellent notebooks on mathematics made by former students which I show my pupils at times to "spur them on" to do better work. After I finish Book IV in geometry, I have them in­ scribe a square in a given circle as accurately as possible on a large piece of wrapping paper. The next day I show all the figures to the class and by tactful suggestion, cor­ rection, and guidance, I secure splendid results. For class work each pupil is sent to the board to.draw a small figure neatly, accurately, and rapidly, and then write the proof in full. They construct the other construction figures in Book IV on large cardboards (24x20 inches). These can be obtained at any bookstore. In order to be able to do work on a large scale, they see that they must do their work accurately, or the errors are very noticeable. For outside work I allow them one week in which to construct University of Texas Bulletin an original design on a large white cardboard. By the neat and accurate figures handed in, I see that the majority have learned the lesson of perseverance, patience and apprecia­tion of the beautiful. An outside person has no idea what extraordinary designs the pupils make by the use of the straight edge and compass. I will endeavor to briefly describe a few of the designs made last term by the students. A boy made a handsome one to be used for a church window. The border consisted of odd patterns of various figures artistically grouped. In the center at a distance was a beautiful cathedral with a tiny star above it. A girl made one of angels ascending into heaven-the head, hair, and flowing robes were con­structed geometrically. Designs for plates were made by others. I can not begin to tell half of the work that was done. After this. is finished, work is begun on the notebooks. I have these in order to give the students concrete and rich courses that bear upon life and give them stimulating views of industry about them. Each day three or four practical problems are assigned, covering a certain portion of the geometry. The pupils dislike repeating work, so I have adopted this method. They voluntarily work the problems given in the book in order to have a better conception as to how to state and work the problems they must originate. The next day the problems are handed in on loose leaf note paper. I correct them. or indicate the places where errors are made. At the end of the term all are handed in corrected. Whenever I find problems unusually distinctive I read them to the class -this spurs the other members on to do better than before. I wish you, who read this, could see the books that have been made. The books are made as follows: . John Smith's book is called Smith's Practical Geometry. They put in a fly leaf, the title page, preface, table of contents, etc., as found in any ordinary text book. The prefaces are very cleverly written, as a rule, each telling what notable contri­bution to the pedagogy of mathematics he has made, and the outstanding features of his book. I tell them that at the Mathematics Teachers' Bulletin end of the term I wish to see whose book is the most orig­inal, and at the same time contains the fewest errors. Now as to the cover designs. Some very original and at­tractive ones have been made. One made hers out of win­dow glass cut the size of a book, and then bound it with red tape. Another pasted tooth-picks on the cover in such a way as to look like raised letters. The insides of the books are uniquely made. Some print the entire book by hand, and then illustrate each problem by pictures cut out of mag­azines, or else hand-painted ones are made. Some are very clever in making drawings and become so enthusiastic that they originate clever pictures for their problems. In arithmetic last year I had practical problems worked in a notebook, as in geometry. This past term I had the pupils make a note-book called "My Home." Each year I try to have some new scheme whereby to interest my pupils, in order to get the best results possible. Now about the home: I obtained books from different sources and had the pupils cut out the pictures of the home they wished. It had to be furnished attractively and at the same time as cheaply as possible. After the fly leaf, the plan of the house was drawn, using a certain scale. Then each day a room was taken up. The cost of papering, or plastering, it was calculated. Then the cost of furnishing each room was estimated by the prices found in the latest catalogues on the subject. Finally they made out the household ac­count setting forth the reasonable expenses for a week for two people. They had to do the "family shopping" for one week, and make a report as to the cost of different articles, and discuss the reason for high prices in certain places compared to others. Then a blank deed was obtained by each student and filled out. Here they got an idea as to how one ought to be made out. Some of the pupils at first had no idea as to the expenses that were incurred in a home, such as light, food, entertainment, charity, clothing, etc. I have only "skimmed" (something I will not let my pupils do) over the work they have done; but I will say this much, on the whole, my pupils have gained increasing strength, have learned to correlate mathematics with other subjects, have begun to reason independently and skilfully by being able to visualize object relations and conditions; are doing better oral and written work, and have improved their study habits. I do not take all the credit for stimulating them, for I am fortunate in having, on the whole, splendid ma­terial with which to work. PAULA HENRY. [Miss Henry is a teacher of mathematics in the Marshall High School.] LOGARITHMS Consider the following series, the first being a geometric progression and the second an arithmetic progression. 1, 4, 16, 64, 256, 1024, 4096, 16384, 65536 0 1 2 3 4 5 6 7 8· These two sets of numbers enable one to perform with ease and readiness a number of operations in multiplica­tion, division, involution and evolution. For example, (a) multiply 16 by 1024. Under 16 is 2 and under 1024 is 5. Add 2 and 5. Over the sum 7 stands 16384 which is the required product. (b) Divide 65536 by 4096. Under the dividend is 8 and under the divisor is 6. Subtract 6 from 8 and over the remainder 2 stands 16, whieh is the quotient sought. (c) Raise 256 to the second power. Under 256 is 4; multiply this by 2 and over the result 8 is 65536, which is the second power of 256. (d) Extract the fourth root of 65536. Under 65536 is 8; divide 8 by 4 and above the quotient 2 is 16, which is the fourth root sought. Other terms may be added to the geometric progression above given by taking the first term and dividing it by 4 and then this result by 4 and so on as far as desired. The arithmetic progression is similarly extended by writing down the natural numbers in descending order. In this manner the following sets of numbers are obtained. 1 1 1 1 - , - , - , -, 1, 4, 16, 64, 256, 1024 256 64 16 4 -4 -3 -2 -1 0 1 2 3 4 5 1 (a) Multiply -by 1024. 64 Add -3 and 5 and above the sum 2 is 16, which is the required product. 1 (b) Divide 4 by -. 256 Subtract -4 from 1 and above the result 5 stands 1024, the required quotient. (c) Raise 14 to the fourth power. Multiply -1 by 4 and over the result -4 is the re­ l quired power -. 256 1 (d) Extract the square root of--. Divide -4 by 2 and 256 1 over -2 is -, the required root. 16 Suppose that the corresponding numbers 16 and 2 in the table were erased. How could they be obtained? The number 16 is the middle term of three successive terms ·in geometric progression of which the first is 4 and the third is 64. Hence it is the geometric mean of 4 and 64. Simi­larly 2 may be obtained by taking the arithmetic mean of 1 and 3. By inserting geometric means between the successive terms of the geometric progression and arithmetic means between the corresponding terms of the arithmetic progres­sion our table may be extended. Doing so we have 1 1 1 1 1 1 1 1 -, -, -, -, -, -, -, -, 1 256 128 64 32 16 8 4 2 -4, -31;2, -3, -21/2, -2, -11/2, -1, -112, 0 * * * * * * * * * * 2 4 8 16 32 64 128 256 512 1024 1h 1 11/2 2 2112 3 31;2 4 41;2 5 1 1 (a) Multiply ­ by-. Under these numbers are -1112 and 32 8 1 -2112; add these and above. the sum -4 is -,the 256 product sought. (b) Divide 32 by 512. Under these numbers are 21/2 and 41/2. Subtract 4112 from 2112 and over the result -2 1 is -, the required quotient. 16 (c) Raise 32 to the second power. Under 32 is 21/2. Multiply this by 2 and over the result 5 is 1024, which is the square of 32. (d) Extract the tenth root of 1024. Under 1024 is 5. Divide 5 by 10 and over the quotient 1/2 is 2, which is the tenth root of 1024. It a.ppears that the table enables one to substitute addi­tion for multiplication, for in every illustration of multipli­cation two numbers of the arithmetic progression are added and above the sum is the product in the terms of the geo­metric series. In the probleJUs of division illustrated, the quotient is found by subtracting the number of the arith­metic progression under the divisor from that in the same series under the dividend and over the. remainder is the quotient. The table makes a problem in involution become one of multiplying two numbers, and a problem of evolution one of dividing one number by another. Logarithm primarily means number of common ratios. The arithmetic progression gives the number of common ratios in the corresponding terms of the geometric progres­sion, i. e. it specifies the number of times 4 must be taken as a factor to produce each of the numbers of the geometric series. The arithmetic series gives the logarithms of the numbers in the geometric series. Regarding the numbers in the first table we may write log 1=0 log 4=1 log 16=2 log 64=3 log 256=4 log 1024=5 log 4096=6, etc. The purpose of a table of logarithms is to substitute addition for multiplication, subtraction for division, multiplication for involution and division for evolution. Any positive real number may be taken as a common ratio or base of a system of logarithms. The most convenient number for base is 10. Logarithms to the base 10 are known as common logarithms. The following table needs no explanation : 1, 10, 100, 1000, 10,000, 1000,000 0, 1, 2, 3, 4, 5. By writing down in succession the natural numbers preced­ing zero and by dividing 1 by 10 and this quotient by 10 and so on, we have · .00001 .0001 .001 .01 .1 -5 -4 -3 -2 -1 so that we have the following common logarithms. · log 1=0 log .1=-1 log 10=1 log .01= -2 log 100=2 log .001= -3 log 1000=3 log .0001= -4 log 10000=4 log .00001= -5 log 100,000=5 This table furnishes material for thought. We may infer for example that as the logarithm of 1 is zero and the log­arithm of 10 is 1, the logarithm of any number between 1 and 10 is more than zero and less than one, in other words it is some fractio::i. Similarly the logarithm of any number between 10 and 100 is l plus a fraction, and the logarithm of a number more than 100 and less than 1000 is 2 plus a fraction. The problem of finding the logarithm of a number which is not an integral power of 10 can be acco~plished by the principle already made use of in extending the table of logarithms to base 4. Insert a geometric mean between two numbers whose log­arithms are known and the arithmetic mean of the loga­rithms of the numbers will be the logarithm of this mean. By :r;epeated application of this principle the logarithm of any number may be found. The geometric mean of two numbers is .the square root of their product and the arith­metic niean of two numbers is one-half their sum. Let it be required to find the logarithm of 7. 7 lies between 1 and 10 whose logarithms are 0 and l, respectively. The geometric mean of 1 and 10 is 3.16228, the arith­metic mean of 0 and 1 is :5. Hence we have 1 3.16228 7 10 0 .5 x 1 The geometric mean of 3.16228 and 10 is 5.62341. The arithmetic mean of .5 and 1 is .75. Hence we have: 1 3.16228 5.62341 7 10 0 .5 .75 x 1 The geometric mean of 5.62431 and 10 is 7.49894. The arithmetic mean of .75 and 1 is .875. Hence we have 1, 3.16228 5.62341 7 7.49894 10 0 .5 .75 x .875 1 The geometric mean of 5.62341 and 7.49894 is 6.49382. The arithmetic mean of .75 and .875 is .8125. By continuing this process of finding the geometric mean of two number~, one less than 7 and the other greater than 7, and of finding the arithmetic mean of the two correspond­ing numbers, we have the following table: Number 1. 10. 3.16228 5.62341 7.49894 6.49382 6.97831 7.23394 7.10497 7.04135 7.00976 6.99402 7.00188 6.99795 6. 99991 7.00090 7.00030 7.00016 7.00004 6.99998 7.00001 6.99999 7.00000 Logarithm 0 1. .5 .75 .875 .8125 .84375 .859375 .851562 .847656 .845703 .844726 .845215 .844971 .845093 .845154 .845123 .845108 .845100 .845096 .845098 .845097 .845098 Problem 2. Find the logarithm of 59. Number Logarithm 10. 1. 100. 2. 31.6228 1.5 56.2341 1.75 74.9894 1.875 64.9382 1.8125 60.4296 1.78125 58.2942 1.765625 59.3523 1.773437 58.8208 1.769531 59.0860 1.771484 58.9533 1.770508 59.0196 1.770996 58.9864 1.770752 59.0030 1.770874 58.9947 1.770813 58.9988 1.770843 59.0009 1.770859 58.9998 1.770851 59.0004 1.770855 59.0001 1.770853 59.0000 1.770852 So far I have considered logarithms as the number of common ratios to be compounded in order to get from unity to any given numbers. Thus to go from unity to 64 the common ratio 4 is compounded 3 times. Logarithms were invented before exponents and negative numbers. In this paper no reference has been made to powers or expo­nents. To connect the old and new points of view take any positive real number a for the base of the system of logarithms and starting with 1 unite the corresponding geo­metric and arithmetic series. 1 1 1 1 -' a• -' as -' a2 -' a 1, a, a2 ' as ' a4, a5 ' -4 -3, -2, -1, 0, 1, 2, 3, 4, 5. The terms of the arithmetic series are the exponents of the terms of the geometric series, and it is at once obvious that the fundamental laws of logarithms are the funda­mental laws of operation in the theory of exponents, and furthermore that logarithms furnish a simple and elegant method of defining negative and irrational exponents. P.H. UNDERWOOD. (Mr. Underwood is the mathematics teacher in Ball High School, Galveston.) A FEW SUGGESTIONS WHICH I HAVE FOUND USEFUL IN TEACHING PLANE GEOMETRY In any line of work there must be a goal if progress is to be made. In geometry it seems to me that the open door to success lies in arousing the student's interest in the sub­ject and it is with this end in view that I use in my own classes devices of the character explained below. It is the purpose of this article in particular to summarize an actual class-room discussion. A certain amount of time must necessarily be spent at the beginning of this course on definitions of geometrical magnitudes. This work may be made more attractive if it is supplemented by actual construction of the figures, in­cluding the simpler constructions, such as bisecting lines and angles, erecting perpendiculars, etc. In this way the student develops the habit of making accurate drawings­a practice which should be required throughout the entire course. The time saved in working the problems from ac­curately constructed figures more than balances the time lost in constructing them, for many truths which otherwise would have been overlooked will be suggested by the drawings. Propositions should, I believe, be taken up in advance of the assignment in the text, especially in the earlier work, in order that the student may analyze the propositions before he sees the synthetic proof. This practice will also avoid memorizing the propositions-one cause of so many fail­ures in geometry. The propostion, "Two triangles are congruent if three sides of the one are equal respectively to three sides of the other", explained below, is an illustra­tion of the method referred to above of proving a new proposition. Class Room Discussion. Two triangles with the sides respectively equal are con­structed and cut out by each student. By questioning, it is found that the triangles cannot be proved to coincide throughout by superposition. For the sake of uniformity, it is suggested that the shortest equal sides be placed to­gether (as in the drawing below). Later another case of · this same proposition may be proved by putting the long­est sides together when one angle is obtuse. The figure as then represented by the two paper triangles is drawn on the board. A summary of previous propositions with reference to proving two triangles congruent will reveal the fact that either two sides and the included angle or two angles and the included side must be found equal, respectively, in each. It is noted that any two homologous sides are equal and that it remains to prove any pair of corresponding angles equal. The line BB" may now be drawn as a hint to the so­lution. This suggests proving the angles at B and B" equal by proving the parts of them equal. In order to find a means of proving the parts equal the "tool box" described below may be used. With this assistance the angles are found to be base angles of two isosceles triangles and there­fore equal in pairs. The whole angles are easily shown to be the sum of the equal parts and consequently equal. Therefore the two triangles are congruent. Now the class is ready to rewrite the proof in the synthetic form which follows: Two triangles are congruent if three sides of the one are equal respectively to three sides of the other. ' , ' , ' / '... . , " ·~ , --~ Given-Triangles ABC and A'B'C' with AB= A'B' AC=A'C' BC=B'C'. To Prove-Triangle ABC=A'B'C'. Construction-Place triangle A'B'C' in position AB"C, letting A'C' coincide with its equal AC and ver­ tex B' fall opposite B. Draw line BB". PROOF Statements Reasons 1. AB=AB" 1. Given. 2. Ll=L2 2. In an isosceles triangle the base angles are equal. 3. BC= B"C 3. Given. 4. L3=L'.4 4. Same as (2). 5. LABC=LAB"C 5. If equals are added to equals the sums are equal. 6. .6ABC=6 AB"C 6. Two triangles are con­gruent if two sides and the included angle of the 011e are equal respectively to two sides and the included angle of the other. 7. 6 ABC=.6A'B'C' 7. Equals may be substi­tuted for equals in an equa­tion. I would like to call particular attention to the form used in the above proof. This type of parallel arrangement of statements and reasons, which is used in some texts, not only gives the student a definite model but also lightens the teacher's burden in grading papers. Frequent written exercises rather than a final test at the end of each month require the student to be prepared and prevent cramming. In order that the papers may be re­turned promptly and at the same time not become a burden to mark, it is advisable that only one problem or proposition be given, to be solved in the few minutes set aside for that purpose. Papers returned with the mistakes marked while the problems are fresh in the student's mind are more bene­ficial than those which are kept overtime or never returned by the teacher. For review work a great deal of time may be saved by having the propositions and problems written on slips of paper before the class period and one given to each student as his boardwork assignment. Another device which Prof. J. W. A. Young calls the "tool box" has become a permanent asset in my classes. The "tool box" contains a set of "tools" grouped according to what they will do. For instance all means of proving angles equal are recorded under one head, those of proving line-segments equal under another head, etc. By adding to this list as the new work is taken up the student will have a convenient reference from which to study out the solutions of new problems. The giving of class grades for special problems handed in has proved to be a stimulant not only to the more progres­sive student but also to the poorer pupil in that the latter will, for the sake of raising his grade, try to solve these problems and so strengthen his work by original thinking. These special problems are of three types, first, solutions not given in the text, second, proofs different from the text and, third, problems a little more difficult than those on which the class as a whole is working. The equipment of the class-room is also a factor in teach­ing geometry. In addition to the usual necessities, a small case for the use of the student would not be out of place. Among other articles it might contain cardboard models of the different figures studied, colored crayons, a protractor, a slide rule with instruction book, several supplementary texts and a History of Geometry. The above methods have been used by me with the hope of awakening the student to the realization that he is not being taught just a collection of facts but is learning to make discoveries for himself. To do this he must study the problems inductively and have the pleasure of finding the solutions for himself. In this way he will acquire the habit of reasoning and thinking independently, which, to my mind, is the greatest value of the study of geometry. MRS. LEE V. BRINTON. [Mrs. Brinton is one of the mathematics teachers in the South End Junior High School, Houston.] "STRATIFIED OR CORRELATED MATHEMATICS­ WHICH?" These words head the advertisement of a new text on Junior High School Mathematics by Miss Gugle of the Co­lumbus, 0., schools. This is a question which high school teachers are being called on to consider and is most certainly one which they should have a part in deciding. It is not new, but vital. We have read with interest the discussions, reports, and results of investigation, which have been set forth within the last few years by the National Committee on Mathematical Requirements, by many prom­inent college professors, and by the various University and State Mathematics Clubs. The National Council of Teachers of Mathematics, organ­ized at the N. E. A. meeting held in Cleveland last spring, together with its control of the Mathematics Teacher through which teachers of mathematics are given a medium of expression, promise to be of invaluable service to teachers of mathematics in secondary schools. The purposes of the Council are given in the January number of this journal, and at the head of the list we find this statement: "We prefer that curriculum studies and reforms and adjustments come from the teachers of mathematics rather than from the educational reformers." I should be very glad to read further expressions from the teachers of mathematics in the Texas schools regarding the curriculum and its content. This is surely no time to fail to take account of the criticisms against mathematics and its teaching. There is now available for the curriculum such a large amount of material, in such a variety of sub­jects, that that which remains must amply justify itself. I beg leave to submit some points favoring the reconstruc­tion of the courses in high school mathematics obtained by the reading and reseach work required for a course in edu­cation at the University of Chicago. The discussion is in outline form, under two heads : Mathematics Teachers' Bulletin A. REASONS FOR CHANGING THE CURRICULUM. a. The reconstruction of the course in high school mathe­matics is a problem of pressing need, owing to the rapid growth and modification of American high schools in re­cent years. b. The present curriculum does not meet the social needs, capacities, and interests of the students. c. It has been shown by actual count that 80 per cent of the problems in the text books in common use are formal, consequently too much time is spent on formal work. d. Unmotivated work is unescapable. e. Algebra, geometry, and trigonometry present peculiar difficulties to boys and girls when these subjects are taught as separate and distinct branches. f. Our method of· teaching algebra first, followed by geometry and trigonometry, is unpedagogical, unhistorical, and accidental. g. Ordinary pupils need the elements of all the subjects, rather than an advanced attainment in any one of them. h. To fill a whole year profitably with either algebra or geometry necessitates carrying the pupil too far into ab­stract and difficult parts too soon after beginning the sub­ject. i. In their present highly organized form it is too diffi­cult to restore to algebra and geometry any close, natural relation to real experience. j. Kriowledge and power are functions of the time, and the successive presentation of the branches, each by itself, with little use for what is learned for long periods, does not get mastered. k. The relatively large per cent of failures and with­drawals in the first year of algebra and geometry prove the course is unfit for the early high school years. I. A program made up of separate one-year units is choppy, incongruous and jolty. m. The current plan is devised for the main purpose of future recognition, for cramming and getting the pupils by the milestones. n. The current plan, intrenched behind the interests of powerful publishing houses, and fortified by the habits of the school masters, is not progressive. B. SUGGESTIONS As TO METHODS OF PROCEDURE. a. Study intensively the psychology of teaching mathe­matics as treated in a text like Judd's, or Dewey's "How We Think." b. Examine and study carefully the latest, best books on methods of teaching high school subjects. c. Base conclusions on scientific experiments. d. Pay more attention to the slow, backward pupil, by personal supervision if teaching in a small high school, by the supervised study period if teaching in a large school. e. Allow for a wider range of individual differences. f. If the text book used is of the formal type, cut out about two-thirds of the formal problems and replace them with the type that will be suited to the needs of the stu­dents. g. Introduce a correlated text-book if possible, for: 1. They are in accord with the mathematical move­ment. 2. They afford many opportunities for broadening and enriching the field. 3. The pupils are introduced into the easier parts of all the subjects first, then those of higher difficulty, final­ly the hardest parts last. 4. The different mathematical gains of the pupil are more easily kept in function. 5. They appeal to more students than algebra or geometry does a!one. 6. Deductions to which geometry is adapted, and the analysis of algebra employed separately and exclusively, do not train the rational faculties as unified mathematics does, particularly in the earlier steps. 7. Pupils are trained to want to know the things rationally. 8. They keep available for the pupils all the time the calculations of arithmetic, the equation and symbolic way of algebra, the proving of geometry. 9. The spirit and morale of mathematics is general­ly improved by correlated mathematics. 10. They will economize time and increase teaching efficiency. 11. They will give the pupil a broader knowledge of the nature of the field of mathematics, so that he is able to see better what lies in store for him if he goes on into the second and third years. 12. If he does not continue he has a better asset from a social standpoint. Mathematics Teachers' Bulletin h. In the first year emphasis should be placed on the equation, formula, graphs, the easier types of factoring, and the simpler notions of trigonometry and surveying. i. In the second year the more difficult t}!eo:rems and originals of geometry, the higher forms of the equation, so­lution of right triangles by natural ·and logarithmic func­tions, trigonometric identities and equations should be in­troduced. j. In regard to continuing the correlated courses through the third and fourth year of the high school, I am not able to speak from experience. But from examining Mr. Bres­lich's Third and Fourth Year Books, I do not see why the idea could not be carried on through the entire course with equally good results. DORA A. WILLINGHAM. (Miss Willingham is the mathematics teacher in the Brownsville High School.) LESS MEMORY WORK AND MORE THOUGHT IN GEOMETRY We are all striving for more thought, though memory work is injurious only because of the pupil's attempt to substitute it for thought. The suggestions I shall give are all borrowed, but have come from many different sources and may be new and of practical value to some teachers start­ing in geometry work. The device that I have found most useful in doing away with the beginner's desire to memorize his book is the use of a note book in which to classify theorems and useful originals. Starting a new subject, such as "Ways To Prove Lines Unequal," they study first the postulates to see what they may take for granted and copy these at the top of their page, then as each new theorem on that subject is proved they add it to the page. Such books can be had with the theorems already classified, but I believe they derive benefit from making their own books . .They use these books on all occasions, for preparing les­sons, for reciting, and-most consoling to them-for taking tests. Since they know they may always have these books, memory work is obviously unnecessary. For thought, we look to their work on the original. The best plan that I know for doing away with a child's aver­sion to originals is to give him, at first, nothing else. I see no reason why a child should have geometry divided into two parts: originals and the better half. I prefer that the beginner use no distinguishing name for them. They are all problems and I hope to see the day when we may use a text with no demonstrated propositions. True, they are harm­less, for you may write any theorem on the board as an orig­inal and they will never discover it in the book unless di­rected to it, and many of them would prefer then to do with­out the book. I usually give them their first "worked out problem" some time in the second month and generally have the gratification of finding that a majority prefer orig­inal work. Mathematics Teachers' Bulletin This way, of course, takes careful development of a theo­rem in class before assigning it as a problem. In other words, never assign work that they are not prepared to do. Always resort to more problems on the old subject rather than plunge them unprepared into a new subject. I would say give them no assignment, rather than one in which they would flounder. Too much of actual study in geometry is done with the idea that "though going nowhere we may inci­dentally get somewhere." Unless they are ready to deter­mine the type of a problem and know how such problems are to be solved they are, as I see it, better off without study until you can direct them. It is interesting to note how often the pupil who comes for help has failed because he has not determined the kind of problem. In fact this question of learning to determine the type of problem needs universal attention. It is not that the pupil cannot solve a quadratic equation when he come to it in geometry, but that he does not know that it is a quad­ratic. I have found construction in connection with other work in geometry a most valuable aid to thought and understand­ing. Instead of taking it as a separate division as in the text. take each construction with the theorem upon which it depends. The time to divide a line into equal parts is un­doubtedly when you learn that three or more parallels cut­ting equal segments on one transversal, will do so on every transversal; and the time to make triangles similar to a given triangle is as you learn the ways to prove them similar. Another means for developing good systematic thinking, and one which is sadly neglected in our texts, is the use of the locus of a point. I have repeatedly heard teachers say that the high school pupil makes little use of the locus of a point. Certainly he does unless he is taught to use it, and the teach­er who is failing to do this is robbing him of a most useful tool. By a constant use of it in class work the teacher can make it of practical value to any child. University of Texas Bulletin Teaching plane geometry is a most conclusive way for the teacher to find what happens to the best laid plans. Still we must have plans and some of these may be of use to others. HELEN CARR. (Miss Carr is Principal of the Orange High School.)