bluuebunny/crossref_metadata_2025 huggingface dataset:
Why Study Mathematics?
Authors: Wren F. L.
Published: January 2021

Why should anyone study mathematics? This type of question is not peculiar to mathematics nor to the field of education. Let us look at a corresponding situation in the field of business. Suppose an automobile salesman attempts to sell a car, what are some of the questions he has to answer? The buyer wants to know the make of the car and compares it with other makes from the standpoint of beauty, service, and economy. Before the sale can be made the salesman must present convincing argument on all of these points and surely no real salesman will attempt such a task without being thoroughly familiar with the car himself. While the analogy may not be complete from the case of the automobile salesman to that of the teacher of mathematics, yet it is certainly true that the teachers of mathematics are primarily the ones who should be able to “sell” mathematics to the “doubting public.” There are two questions that every mathematics teacher should be able to answer if be is to be able to give an intelligent answer to the one already proposed : they are “What is mathematics?” and “What relation does mathematics have to the cultural, industrial, and recreational activities of a progressive civilization?”
The Problem of the Teaching of Exponents
Authors: James Lyons William
Published: January 2021

Though the problems arising in the teaching of exponents cannot be said to be insignificant, there has been little written on the subject, as compared to the material on the teaching of other topics in the algebra course. We will not here seek to define a reason for the dearth of material on the teaching of exponents. Suffice it to say that the task of teaching exponents so that they will be meaningful and understandable to the pupils presents difficulties. The writer has collected suggestions from artides and works on the teaching of secondary mathematics, and has analyzed the treatment of exponents in several textbooks. In this paper he has endeavored to choose and refine these various suggestions so as to produce a single, unified method of teaching exponents.
Problems of Algebra Pupils
Authors: D. Murthy Clara
Published: January 2021

This topic was chosen because I have the feeling that algebra teachers in general are not as conscious as they should be of the problems that confront their pupils. As they teach the same subject matter year after year, its difficulties become less apparent to them and the tendency is to forget that the ground is all new and more or less rough for their pupils. Before I began to teach, Professor Burrell of Wellesley College gave me this advice, “Always keep on hand a piece of work that is so difficult for you that you have sympathy for your pupils and understanding of the difficulties which they are encountering.” I have not always followed this advice, but I believe in it.
Club Topic: Symmetry
Authors: Anning Norman
Published: January 2021

The student who is preparing to discuss such a topic in mathematics club will find plenty of suggestions in F. C. Boon's Companion to Elementary School Mathematics, Longmans, Green and Co., 1924. Mention is made, as it is in any good textbook, of the bilateral symmetry of the coefficients of the expansion of (a+b)n where n is any positive integer. It is the purpose of this note to suggest that this fact is a member of a hierarchy of facts which an interested student would find easy to explore and to explain. The restriction of the exponent n to positive integers remains throughout.
Mathematics Journals
Authors: Smith Wallace
Published: January 2021

Because of the interest voiced by the teachers of mathematics at the recent meeting of the West Virginia Council of Mathematics during the S.E.A., I am induced to comment on the value of two bulletins published by and for mathematics teachers.
A Psychological Explanation of Failures in High-School Mathematics
Authors: H. Judd Charles
Published: January 2021

This paper is a plea by a psychologist for the treatment of students who are having difficulty in mathematics by the same kind of skillful, scientific methods as are employed in dealing with human beings in need of physical relief. I begin by accusing you of not always being efficient in your work. I have it in mind to present some statistics that I feel sure ought to be of interest to you. I find these statistics one of the best introductions to my talk because I find they always secure attention for what I have to say, when otherwise I should not be able to command attention.
A Maintenance Program for Tenth Grade Mathematics
Authors: Pruitt C. C.
Published: January 2021

Probably no subject in the high school curriculum is receiving more attention today than that of plane geometry in the tenth grade. Much of this attention is directed towards the possibility of fusing plane and solid geometry into one course. From this situaation, one would infer that all is not well in either the field of plane geometry or that of solid, with probability in both. I think all teachers of mathematics in the senior high school are agreed that the teaching of plane geometry has not advanced to the point where we are satisfied with the results obtained.
Curriculum Adjustments
Authors: Meyers W. C.
Published: January 2021

Before considering the subject “curriculum adjustments” as they apply to slow moving pupils, it will perhaps be worth while briefly to call attention to some historical facts regarding the growth of mathematics, in the high school and college courses. The teachers of mathematics, like many school officials, are inclined to assume that the mathematical content we are teaching today is the same as that taught by our forefathers. This is a false assumption: quite the contrary is true. Our present mathematical curriculum bas been with us hardly long enough to have a tradition, certainly not long enough to have acquired a sacredness which must be worshipped.
The Contribution of High School Teaching to Ineffective College Teaching
Authors: Seidlin Joseph
Published: January 2021

In a Recent completed study on the teaching of college mathematics† I reached the almost inevitable conclusion that, barring exceptions, college teaching is generally ineffective in the sense that
Kenneth Grahame
Authors: Gray James
Published: January 2021

It is difficult to believe that Kenneth Grahame who died the other day at his home in Pangbourne, England, was a man of 73. Though his books for children called The Golden Age and The Wind in the Willows were published before men now in their thirties were born, still it is a shock to know that their author has grown old in the meantime. One unreasonably expects the perennially youthful quality of the books to have kept the man eternally young.
Real Business in the Eighth Grade
Authors: Brown Christine A.
Published: January 2021

“Why can't we have a store for the accommodation of our school? It would be much more convenient to buy books and school supplies right here in the building. And what fun it would be!”
Demonstrative Geometry for the Ninth Grade
Authors: Reeve W. D.
Published: January 2021

Why Demonstrative Geometry in the Junior High School? If the pupil is ever to gain a knowledge of what it actually means to prove something and he has to leave school by the end of the ninth grade, he ought to be given a chance to study demonstrative geometry while he is still in school. We know that some of our greatest mathematicians were not awakened to the beauty of the subject until they studied geometry. Sir Percy Nunn once told me that he got his first real thrili when he came to the theorem about the sum of the angles of a triangle. It is related that Newton showed little promise until he took up the study of geometry.
The Ideal Preparation of a Teacher of Secondary Mathematics from the Point of View of an Educationist
Authors: Bagley William C.
Published: January 2021

It would scarcely be profitable, I think, to place too strict a construction on the term “ideal” as used in the title of this paper. In fact I have been so continuously confronted with the handicapping conditions that must be met in preparing teachers that I should hardly know what to do or to say if these conditions were suddenly removed. For example, there is always the matter of the time that can be given to preparation; there is always the type of person whom we may expect to make the preparation; there are always large gaps in our knowledge as to the best ways in which to select and organize materials; and in the preparation of a highschool teacher there is the practical need of providing training that will enable him to teach in at least two fields of specialization.
Report of the Policy Committee
Authors: Hassler J. O.
Published: January 2021

A brief statement of the purpose of the Policy Committee was printed in this journal in the issue of November, 1932, under the title, “A Tentative Program of the Policy Committee.”
Colin Maclaurin
Authors: Sanford Vera
Published: January 2021

Colin maclaurin seems to have been a prodigy in mathematics. He entered the University of Glasgow at the age of eleven and took his masters degree at fifteen. At nineteen, he earned the post of professor of mathematics at Aberdeen by competitive examination. He spent several years in travel as tutor to an English family. In 1724 he won a prize offered by the French Academy of Sciences with an essay on the percussion of bodies. In 1740, he divided a similar prize with Euler and Daniel Bernoulli, his entry being an essay on tides. In 1725 he was made the deputy of the mathematics professor at Edinburgh and later became professor of mathematics there. The Edinburgh appointment was one in which Newton was greatly interested and Ball says that “Newton privately wrote offering to bear the cost so as to enable the university to secure the services of Maclaurin.”*
Development of Mathematics in Secondary Schools of the United States
Authors: Wren F. L. , McDonough H. B.
Published: January 2021

The Latin grammar schools had inherited certain aristocratic characteristics that existed in England at the time the American colonies were settled; these characteristics exhibited themselves in the practice of selecting pupils according to the rank and social status of their parents. From the conditions incident to frontier life they had acquired certain other characteristics such as limited means, limited facilities, limited purposes, and limited opportunities, all of which were manifested in the narrow curriculum of the period. As the population of these early colonies increased through immigration and birth, old communities broke up and migration westward began. The new settlements established in the wilderness were founded by people who had not known the religious zeal and oppression of the old country. With this shifting of population new interests in shipping and commerce began to replace the old interests in religion and agriculture. Such social and commerical expansion gradually led to a demand for a more liberal and democratic form of education, which demand was met by the organization of the academy. This new institution, to a certain extent an offspring of Philistinism, “came in to serve the broader need represented by those who would enter occupational pursuits without going to college, as well as those planning to continue their education. It was, in effect, an expression of expanding democracy.”1
Mathematics in the Integrated Curriculum
Authors: Zant James H.
Published: January 2021

This paper very properly begins with a definition. Integration, as we all know, means to make into a whole, or to combine into a completely related unit. From the standpoint of a school program it means that the student not only has learned many things in school but he should also have acquired a unified idea of all the parts and of their relations to one another and to topics outside. It is an aim of education and a worthy one. Teachers everywhere hope that the individuals whom they teach will eventually be able to react to most life situations with intelligence and skill.
The Tree of Knowledge
Authors: Reeve W. D.
Published: January 2021

In the Hall of Science of the Century of Progress Exhibition last year in Chicago there was represented on one of the walls the “Tree of Knowledge,” a photograph1 of which we present (by permission) on page four of this issue of The Mathematics Teacher. As can be seen from this picture mathematics furnishes the central root and vitalizing energy for the basic sciences such as astronomy, botany, chemistry, geology, and physics and, together with them, furnishes strong support for the applied sciences on the higher branches such as social studies like economics and sociology and engineering of various kinds.
The Need for Reorientation of Mathematics in the Secondary School
Authors: Hutson P. W.
Published: January 2021

Any general shift in educational theory necessitates a general readjustment of educational practice. That is axiomatic. If our canvass of the need for reorientation of mathematics suggests the abandonment of long-cherished conceptions of value or method and the acceptance of unfamiliar goals and means of attaining them there may be some comfort in the realization that the specialists in other fields of learning are facing similar problems of adjustment. It is my conviction that all the subject fields of secondary education can be considered from the angle of the topic of this afternoon's discussion quite as profitably as mathematics.
The Calculation of the Approximate Rate of Interest in Consumer Credit
Authors: Kinney Lucien B.
Published: January 2021

As one phase, perhaps, of the interest that has been developed in consumer education, some of the more familiar aspects of consumer credit are beginning to receive attention both in junior high school mathematics and in commercial mathematics. Many of the modern text books in these subjects include topics on installment buying and building and loan associations, and it is to be expected that the general interest in consumer credit will lead eventually to a rounded treatment so far as this is possible on the secondary school level.
News Notes
Authors: William C. Krathwohl
Published: January 2021

The National Council of Teachers of Mathematics held its First Summer Meeting at Denver, Colorado from June 29 to July 1, 1935.
New format For the Mathematics Teacher
Authors: Reeve W. D.
Published: January 2021

With this issue, The Mathematics Teacher is changed in form and in size. The new format enables us to give our readers more material and to present it in a more desirable way. This is done by increasing the size of the page so as to permit two columns of printed matter to appear. This double column effect makes the reading process easier and the artistic effect better. Along with this change in format, it is hoped that we shall be able to introduce other new features that will make the magazine more attractive and helpful to teachers of mathematics. What we can ultimately do to improve the magazine will depend largely upon the financial support the National Council of Teachers of Mathematics can get from those in whose interest the magazine is published eight times each year. With renewed efforts on the part of our present membership we should be able to secure many new subscribers. No meeting of mathematics teachers should be held without someone's taking the responsibility of presenting the case for the Mathematics Teacher. Subscription blanks for use at such meetings like the one appearing as the first advertisement of t his issue can be had postpaid for the asking. The blanks also give a description of the National Council Year-books which now constitute an essential part of the library of many teachers of mathematics. We shall be glad to have the comments of our readers on the new format and also any suggestions for the fur ther improvement of the magazine along any particular line.
What Day of the Week Was It?
Authors: Canaday E. F.
Published: January 2021

To find the day of the week on which any given date fell, add four easily found numbers, divide by seven and the remainder is the day of the week desired. With practice the process will take only from five to twenty seconds. For example, for June 5, 1922, we have 1 + 6 + 4 + 5 = 16 = 2.7 + 2. The remainder being 2 the day was Monday.
A Message From the New President
Authors: Hildebrandt Martha
Published: January 2021

When you and I began this work in which we are now engaged, we pledged ourselves consciously or unconsciously, to help those who came within the influence of our classrooms to become the highest type of individuals of which they were capable; to become intelligent citizens. We wished for them a development which would enable them to adjust themselves to the civilization of their time, a development which would make them citizens with an interest in and a knowledge of the forces which shape and control their environment. Mathematics has and is definitely contributing to this environment and so it must assume its fair share of the development of our pupils.
A Protest Against Informal Reasoning as an Approach to Demonstrative Geometry
Authors: Hendrix Gertrude
Published: January 2021

Three stages are commonly recognized in current recommendations for a secondary geometry course: (1) the intuitive stage, (2) the stage of informal reasoning,1 and (3) the stage of formal reasoning, or demonstrative geometry. The second stage does not provide the transition to the third that is desirable. In fact, that informal reasoning has any value at all as an approach to the formal reasoning stage is questionable. An attempt to present reasons for doubts is set forth with care in the succeeding pages of this discussion. If the conclusion to which these reasons lead is valid, a decided change in the present methods of approach to demonstrative geometry should be effected.
The Art of Teaching A New Department
Authors: Sweedler Esther
Published: January 2021

The student studying physics, biology and chemistry can apply and test his text book theories in the laboratory but the mathematics pupil learns his rules and formulas without ever being given the opportunity to apply them. I recall with some amusement now my embarrassment when a youngster at a summer camp asked me how I would go about finding the distance across the camp's lake. I had just completed a major in college mathematics but the simple procedure necessary to find the distance across the lake was very vague in my mind.
On the Curriculum of Mathematics
Authors: Smith Eugene R. , Rorer Jonathan T. , Schwatt Isaac J.
Published: January 2021

The student must thoroughly understand the meaning and the philosophy, so to speak, of each mathematical concept presented to him. It takes a long time before he familiarizes himself so thoroughly with the conceptions of any mathematical subject so that he gets their significance, meaning and spirit, and the ability and facility to apply them readily. Few students have a clear understanding of the quantitative meaning and significance of the theorems of proportion, such as: “A line drawn parallel to a side of a triangle divides the other sides proportionally,” or, “Similar triangles are to each other as the squares of the homologous sides.” In all my experience I have not received from a pupil a satisfactory explanation of the truth that one divided by infinity is equal to zero, a conception used in secondary mathematics. The same is true of college mathematics. Few students who have studied Analytical Geometry are able to give the true meaning, for instance, of the equation of the straight line, y = sx + m, i. e., that the ordinate of any point of the straight line is m greater than s times the abscissa of the point. These instances may be multiplied to quite an extent.
High Common Factor
Authors: Hart Howard F.
Published: January 2021

As the colleges now are requiring highest common factor by factoring methods only, any plan whereby the number of factors to be tried can be lessened is certainly worth while. For in general we must regard as possible any binomial factor whose first-degree term is an integral divisor of the highest term of the given expression and whose independent term is an integral divisor of the independent term of the expression. Thus in such a problem as, “Find the H.C.F. of 5x3 − 21x2 + 5x − 4 and 5x3 − 19x2 + 5x + 4” (McCurdy’s Exercise Book, page 40, example 4) the possible factors that a student might try and must try, unless he were very lucky in those he chose to try first, are x ± 1, x ± 2, x ± 4, 5x ± 1, 5x ± 2, 5x ± 4. And further if the given expressions were, say, cubics having no common binomial factor at all but with a quadratic one instead (e. g., 2x3 + 5x2 + x − 3 and 2x3 − x2 − 5x + 3) I doubt if the ordinary first-year student would get any result unless it were unity.
Let's Face the Facts
Authors: Douglass Harl R.
Published: January 2021

For a generation there has been con siderable ferment with respect to the place and content of mathematics in the high school curriculum. Two central issues have been prominent: (1) Does mathematics as now taught constitute a more suitable content for the education of the great mass of high school pupils than other subject matter which might be substituted in its place? (2) Should thecontentof high school mathematics be thoroughly re-organized?
Is Mathematics Teaching on the Secondary Level on the Spot?
Authors: Klopp W. J.
Published: January 2021

In beginning the 1935 and 1936 series of Educational Symposia on the secondary level, for the various departments, it occuned to me that it would be quite in harmony with trends in progressive education, to have each department take for its major theses for discussion, certain definite outcomes which we have a right to expect as a result of experiences provided by the secondary schools.
The Art of Teaching A New Department
Authors: Scott Nell
Published: January 2021

The date for the observance of National Foreign Trade Week had been announced. In reply to a letter addressed to the Los Angeles Harbor Commission we received their last Annual Report, 1934–1935; their previous Annual Report, 1933–1934; and their Monthly Statistical Report for June, 1935. These Reports were placed in the hands of two seventh grade mathematics classes, in a junior high school, for the study of domestic and foreign trade over the wharves of the Los Angeles Harbor.
Mathematics and the Integrated Program in Secondary Schools
Authors: Reeve W. D.
Published: January 2021

In the Journal of the National Education Association for May, 1934, there appeared on page 137 the following comment on the “trend toward integration” in American secondary schools:
Individualized Instruction in Algebra
Authors: Snader Daniel W.
Published: January 2021

It has been said, time after time, that the most neglected pupil in our high school is the brilliant boy or girl. There seems to be a leveling down of the classwork to meet the “tempo” of the average child and partially, if not wholly, neglect the superior students upon whom our future leadership will depend.
The Art of Teaching A New Department
Authors: Adkins J. B.
Published: January 2021

This lesson was used with ninth grade classes in elementary algebra. Previous work with graphs included the usual statistical graphs and a few formulas of the types d = 8h and s = 6w + 10.
Random Notes on Geometry Teaching
Authors: Barber Harry C.
Published: January 2021

There has been much discussion of what should be done with the superposition proofs. The plan often advocated is to postulate all the theorems usually so proved. In England the reason given is that when we move a figure we cannot be certain that it does not change in size and shape. In the United States the reasons given are that superposition is not a suitable method at the outset; it is not readily understood by the beginner, and it interferes with the later use of other methods of proof.
Helping Mathematics with an Exhibit
Authors: Krathwohl William C.
Published: January 2021

This is a story of how one city is trying to overcome the prejudice which exists in some quarters against mathematics.
The Teacher's Professional Study of Subject-Matter in Mathematics
Authors: Bagley William C.
Published: January 2021

I am somewhat handicapped in dealing with the subject assigned to me on the program since I discussed certain phases of it in a paper that I presented before this Council in 1933.1 While few if any of you will recall that paper, it is a matter of record in your journal and I dislike under these conditions to repeat what I have already said. I may be permitted, then, to outline very briefly here some of the considerations that led to the development of the conception of the professional treatment of subject-matter as a central feature in the education of teachers, and to set forth some of the implications of this conception as applied to the preparation of those who are to teach mathematics as a phase of general rather than of specialized education.
The Art of Teaching: The Laws of Signs in Multiplication
Authors: Zant James H.
Published: January 2021

In a recent issue of The Mathematics Teacher1 a method of teaching the law of signs in multiplication was suggested and defended. The suggested procedure was essentially as follows: in the first type problem (3×5) the pupil was led to formulate the fundamental definition of multiplication by a positive whole number as a process of finding the sum of three equal groups; in like manner the second example (3 × − 5) is disposed of, that is, the pupil obtains the answer by adding three minus fives. In neither case, however, is an attempt made to get the pupil to formulate an explicit general definition of multiplication by a positive whole number.
Vitalizing Geometry with Visual Aids
Authors: Drake Richard , Johnson Donovan
Published: January 2021

Geometry more than any other subject is dependent upon visual aids. Every theorem and every exercise requires a diagram in a book, on the blackboard or in one's mjnd. Yet probably in no subject have teachers shown less progress in developing visual aids and improved visual techniques. Newer textbooks have increased the number of photographs of church windows, bridges and decorative patterns, but much more visual material is needed to bring geometry closer to everyday things. The mathematics exhibit at the world's fair in Chicago indicates how interesting mathematics can be made when it is presented in concrete terms rather than in the usual abstract manner.
The Trisector of Amadori
Authors: Daniells Marian E.
Published: January 2021

It is well known that, alt hough the Greeks searched unsuccessfully for a method of trisecting an angle by means of an unmarked straight edge and a compass only, they did devise methods of trisecting by other means. One of the oldest of these methods, which is attributed to Archimedes was known as a “veoσis” or verging” and consisted essentially of making a line approach a specified length or position.
Presenting the Report of the Joint Commission to the National Council of Teachers of Mathematics
Authors: Breslich E. R.
Published: January 2021

The Development of the mathematical curriculum of the secondary schools is divided into two periods, one from 1893 to 1923 and the other from 1923 to 1940. The first period begins with the report of the famous Committee of Ten on Secondary Studies, the second with the report of the equally famous Committee on Mathematical Requirements. Each report has made a lasting impression on the mathematical curriculum. The presentation of the report of the Joint Commission marks the beginning of a third period in the advancement of the course of secondary school mathematics. Its influence will soon be seen in the forthcorrung textbooks and in the newer courses of study.
Recent Trends in Arithmetic
Authors: Sueltz Ben A.
Published: January 2021

All of us are aware of recent trends and developments in the field of arithmetic. For convenience, the more important of these will be discussed under the following headings: Philosophy and Curriculum, Testing and Evaluation, Research, Teaching, and School Practice. After the preliminary presentation, we may wish to discuss more fully several of the topics about which questions are raised.
The Use of Applications for Instructional Purposes
Authors: Olds Edwin G.
Published: January 2021

Introduction. As a group, teachers of mathematics are respected and admired for their skill and interest in communicating knowledge of their subject. This universal recognition of their high standards of competence is well deserved. There seems to be no other field in which the experts are more willing to examine their knowledge of subject matter objectively and to bend every effort to remedy any weakness perceived. I doubt whether there is another subject in which the instruction is less open to criticism because of violation of pedagogical principles. Having attained leadership by insistence on thorough knowledge and by intelligent attention to instructional techniques, mathematics teachers can be depended upon to use the same qualities in maintaining that leadership.
Looking at Solid Geometry Through Perspective
Authors: Spearman Ethel
Published: January 2021

“Much attention should be given to the visualization of spatial figures and relations, to the representation of three-dimensional figures on paper, and to the solution of problems in mensuration,” says the Joint Commission among its suggestions on solid geometry in its report on “The Place of Mathematics in Secondary Education.”* These ideas are incorporated in the notion of perspective. Perspective may be regarded as a practical means for securing a rigorous reciprocal metrical relationship between the shapes of objects as definitely located in space and their pictorial representation. It may be regarded as the rationalization of sight. This Commission further points out that “the problems in mensuration offer opportunity for correlation of solid geometry with arithmetic, algebra, and trigonometry. Part of the importance of the geometry of the sphere comes from the perspective that it makes possible.”
What Is Topology?
Authors: Hall Francis C.
Published: January 2021

A good part of the study of geometry in high schools hinges on two ideas: that of congruence of lines and that of congruence of angles. In the study of projective geometry, a basic idea is that of perspectivity. For example, the points A1A2A3 are said to be “perspectively related” to B1B2B3 A1B1,A2B2,A3B3 meet in a common point P.
A Report of the Seventh December Meeting of the National Council of Teachers of Mathematics at Baton Rouge, Louisiana
Authors: Charlesworth H. W.
Published: January 2021

This convention was held jointly with the Mathematics Association of America and the American Mathematical Society on December 30, 1940-January 1, 1941. 189 registered and about 350 attended.
Mathematics and the Defense Program
Authors: Bacon H. M.
Published: January 2021

As we are all aware, the Defense Program consists primarily of two main efforts: industrial and military. Some would add a third, the effort for political effect. But actual preparations for defense involve first, the training and organization of army, navy, and coordinate services, and second the supply of materiel and the organization of industry and industrial personnel to provide the necessary supplies for the use of the defense services. An elaborate analysis would subdivide the program into a vast number of subsidiary programs, but unless we confine ourselves to a quite general discussion of how mathematics is related to these two principal problems—industrial and military preparedness—we shall be here until late this evening, or perhaps I should say that the speaker would still be talking if he knew enough about the subject, but the audience would long since have sought the freedom of the outer air.
Mathematics of Aviation
Authors: Beito Edwin A.
Published: January 2021

The rapid development of aviation in this country and the great interest shown by our young people should be a challenge to all teachers to make the most of its educational possibilities. The United States government has established pilot training schools in every section of the country, which are open to both college and noncollege students. Thousands of private pilots are now being trained, and the future program calls for lhe training of 30,000 pilots per year. The Air Youth of America, an organization sponsored by the National Aeronautics Association and Winthrop Rockefeller, is forming Model Airplane classes in every city in the country, and the membership is increasing very rapidly. In my own city of Wichita, Kansas, more than five hundred boys and girls have joined the classes. These classes are held in connection with the public schools, after school hours.
Scope of Mathematical Offerings in Selected Junior Colleges
Authors: Ahern Lorella
Published: January 2021

Perhaps no one of those subjects which serve bolh the prepa ratory and terminal functions has as great a range of possible channels by which to serve these functions as has mathematics. Even a cursory enumeration of the avenues by which muthematics may fulfill many terminul and preparatory functions causes one to be confronted by two questions. Is the junior college including the numerous preparatory and terminal accomplishments that are embodied in the mathematical fields? If not, are many of these opportunities being neglected because of a lethargic attitude displayed by mathematics departments which proffer a potentially extensive subject matter in a narrow traditional manner?
The Art Of Teaching: An Experiment in the Use of Graph Papers
Authors: Sleight Norma
Published: January 2021

Representing statistical data by means of the graph plays such an important role in our present day world that it behooves us as the teachers of mathematics to investigate with our classes the uses and misuses of this powerful tool. The pupils come to their junior year well acquainted with the procedures for making simple statistical graphs but with little knowledge of their real uses. The interpretation of circular graphs is very clear to them because they are simple and they have seen so many of them in various publications, but the possibilities of the line graph seem to be vague. They interpolate where they shouldn't, and extrapolat ion is a new word to them.
Mathematics in Our Schools and Its Contribution to War
Authors: Levy Sophia H.
Published: January 2021

Less and less mathematics has been taught in our schools dming recent years; fewer courses have been given than formerly, and subject matter in courses remaining has been diluted until at times the entire character of the material presented has been changed. Meanwhile we have heard that courses in mathematics must give way to courses in social studies of one kind or another. The outlook for mathematics has been growing darker and darker during recent years.
High School Mathematics in Artillery Fire
Authors: Bacon H. M.
Published: January 2021

It is becoming increasingly evident that sound knowledge of the mathematics usually taught in the elementary and secondary schools is an essential part of the equipment of large numbers of men and women who wish to take their proper places in our present day world. This is a fact which could have been established before the questions of Defense and War efforts arose. But the demand for men and women to do industrial work in connection with the war, and for properly prepared personnel by the armed Services, has served to call the importance of mathematics strikingly to our attention.
The Art of Teaching: Vocabulary in Plane Geometry
Authors: Keesler Earl R.
Published: January 2021

Since the technical vocabulary of geometry is composed largely of derivatives from Latin and Greek words, the teaching of their history together with pronunciation, meaning, spelling and use should be included as an important part of the course. English equivalents make it much easier to remember definitions which otherwise may be memorized mechanically. For this reason it is sound practice to discuss derivations whenever they may aid in the assimilation of new words. The Latin and history departments are always more than willing to cooperate in such an integration project, while the increased interest and understanding are well worth the extra effort and planning.
The Art of Teaching: Teaching Solid Geometry
Authors: Wylie Nancy C.
Published: January 2021

This Article is corroborating and supplementing the timely suggestions made by James V. Bernardo in the January 1940 issue of The Mathematics Teacher, on the teaching of solid geometry. I say, timely; first, because of its contribution to the meager body of material on the teaching of these books of Euclidean geometry; second, because most instructors are ready to begin a new semester and desire all additional light on methods of presenting the three-dimensional concepts. Any aids for perfecting technique that will enable the instructor to help the pupil in developing his spatial imagination will, very probably, be received with enthusiasm.
Basic Mathematics
Authors: Mansfield Ralph
Published: January 2021

During the past year, a continual stream of books has been appearing on the desks of many teachers of mathematics. Almost without exception, these books purport to expound “basic mathematics,” to be taught to students who will one day become members of the armed forces of the United States.
The Art of Teaching: On Teaching Percentage
Authors: Bucklen Harry E.
Published: January 2021

When planning to become acquainted with the nature and use of percentage, the eighth grade students of the Bemidji State Teachers College Laboratory School, with the guidance of a student teacher, decided to set up a number of individual business establishments as a framework for life experiences.
Mathematics in the War Program in the Secondary Schools
Authors: Orleans Joseph B.
Published: January 2021

At a convention of Deans last Fall in Albany, N. Y., one of the speakers, I am told, in discussing the guidance of pupils in these times of emergency, proposed as a slogan, “Praise the Lord and pass your mathematics.” Whether she said this in a facetious vein or not, the comment points to a growing trend in secondary school guidance at least for the duration.
Mathematics in Our Schools and Its Contribution to War
Authors: Levy Sophia H.
Published: January 2021

Until a year ago, statements that our high school graduates could not do arithmetic were dismissed as of no consequence, in fact, were almost not believed. But Admiral Nimitz's letter concerning the failures in arithmetic tests given recruits entering the Navy, coming as it did at the very beginning of the War, got people more “arithmetic minded” in a few weeks than had all the efforts of teachers of mathematics in our secondary schools and colleges in nearly twenty years. Suddenly arithmetic has been revived. Suddenly there has been a large increase in the number taking courses in mathematics in the secondary schools. Suddenly there has been a large increase in the number taking courses in secondary mathematics at the University. During the semester now closing we have had 3600 students in our department as against 3000 one year ago. This is an increase of 20%, though our University enrollment dropped from 15,000 to 11,000, and enrollment in advanced courses in mathematics suffered a large drop during the same interval. We have had 1100 people taking courses in secondary mathematics. We have had 500 taking intermediate algebra with background of but one year of elementary algebra.
Horner's Method and the Algorithm for the Extraction of Square Roots and Cube Roots
Authors: Conkwright Nelson B.
Published: January 2021

It seems not to be generally known that the algorithms for the computation of the square root and the cube root of a number N differ from Horner's procedure for the solution of the equations x2 — N = 0 and x3 — N = 0 only in the formal arrangement of the work. It is the purpose of this discussion to show how the resemblance between the two methods of computation can be utilized in the extraction of roots by means of the algorithm.
A General Method for the Construction of a Mechanical Inversor
Authors: Ahrendt M. H.
Published: January 2021

In his book, Linkages: The Different Forms and Uses of Articulated Links, DeRoos stated the following theorem: “In any quadrilateral whose diagonals may intersect at an angle, a point A of a diagonal can always be taken so related to two of the sides that O, A, and B shall be in a line and so that the product of OA and BA shall be a constant.” DeRoos did nothing more in his book, however, than merely state the theorem. He offered no proof of its truth and gave no specifications for the construction of such a linkage. In searching through the literature on linkages, the writer has been unable to find any further mention of this theorem.
The Old Time Religion
Authors: Nason Evan A.
Published: January 2021

In order to furnish a background for what I have to say this morning, may I ask you to go back with me to the period of American History sometimes called the “Roaring Twenties.” This decade from 1920 to 1929 was a fantastic period in our history. It was a period of fads and fancies; a time of revolution in dress and manners; a period of prosperity, easy money and unconventional thinking and living. This was the decade of mahjong and crossword puzzles; of Coué and his magic formula “Day by day in every way I am getting better and better.” Vitamins had not yet been discovered but prohibition was in effect and deficiencies in vitamins A, B, C and D were more than compensated by liberal doses of wood alcohol.
An Open Letter to Teachers of Mathematics
Authors: Withers A. M.
Published: January 2021

An important article by Professor Robert A. Hume, “Shall We Be More Practical?” School and Society, January 2, 1943, links together (not for the first time, to be sure) the fortunes of the pure mathematics in the high schools and colleges with those of the ancient and modern languages. We language advocates must never forget, he says in effect, that when educationists swing their cudgels at the heads of algebra and geometry, our own heads are in danger; and by the same token when they “crack down” upon the languages, algebra and geometry and all their kith and kin had better look to themselves. You have yourself no doubt noticed that the writers who belabor high-school mathematics in the lay and professional prints often couple the foreign languages with them as participes criminis.
Editorial
Authors: Smith David Eugene
Published: January 2021

David Eugene Smith, professor emeritus of mathematics at Teachers College, Columbia University since 1926, died at his home in New York City on Saturday, July 29, 1944, at the age of eighty-four.
The Art of Teaching: A Technique for Teaching Subtraction of Signed Numbers
Authors: Romig W. E.
Published: January 2021

One of the chief obstacles in the way of teaching subtraction of signed numbers is the difficulty of leading the pupil to a proper conception of “opposites.” This difficulty arises, no doubt, from the fact that there are two different conceptions of opposition involved in algebraic subtraction, (1) opposition of value, and (2) opposition of operation.
The Mechanism of Gravitational Force
Authors: Watson Emery E.
Published: January 2021

Although it has been known since the time of Newton, nearly three centuries ago, that bodies attract each other according to the inverse-square law, no explanation as to how gravitation operates has been accepted. However, with our present knowledge of matter it would seem that some explanation should be possible. Keeping in mind Newton's first law of motion; namely, “Every body continues in its state of rest or uniform motion in a straight line unless compelled to alter that state by impressed force,” let us consider, for the purpose of illustration, an atom within which the motion is produced by the vibration of the nucleus in whole or in part, or by the electron as it rotates on its axis or as it travels in its orbit around the nucleus. The direction of rotation or revolution of the particles within the atom is apparently unimportant since the nucleus of the atom may rotate with or in opposition to the motion of the electron as is evidenced by the position of the spectrum lines.
The Trigtractor—A Visual Aid for Teaching Trigonometry
Authors: Eagle Edwin
Published: January 2021

The use of the word, function, in connection with the trigonometric ratios, implies related change. It is difficult, by means of a picture, or even by a series of still pictures, to develop in the pupil's mind this idea of functional change relationship. Failure to develop this concept adequately is the basic cause of very much of the difficulties which arise throughout a trigonometry course. To many, “sin A equals opposite side over hypotenuse, or ordinate over distance,” is a verbal memorization without a clear mental picture, and with the idea of functional dependence entirely lacking. To meet this difficulty in his trigonometry classes at the University of California, Dr. Merton Hill uses a large protractor with a movable radius strip carrying a movable ordinate strip at its outer extremity. The visual aid device pictured and described in this article, which for lack of a better term I call a “trigtractor,” is a further development of this protractor idea. It is a real time saver and simplifies matters much in: (1) teaching clear concepts of the trigonometric functions; (2) representing functions as a single line; (3) explaining changes in the functions from positive to negative and vice-versa from quadrant to quadrant; and (4) comparing functions of angles in one quadrant with those in another, especially in expressing functions of obtuse and reflex angles in terms of functions of acute angles.
The Art of Teaching: Some More Thoughts on Placing the Decimal Point in Quotients
Authors: Johnson J. T.
Published: January 2021

The article on Some Thoughts on Placing the Decimal Point in Quotients which appeared in the February issue of the Teacher calls for a continuation of some more thoughts as it leaves the reader somewhat in doubt and it does not give all of the weaknesses of the so-called logical method.
Some Thoughts on Tomorrow's Mathematics
Authors: Berman S. L.
Published: January 2021

How times have changed! A number of years ago, the educator who espoused the cause of increased mathematical study for secondary school pupils would have been tolerated in some quarters, considered eccentric in others, but would have been ignored completely or not too quietly ridiculed in most educational circles. Now, not only are schoolmen deeply interested in the extension of mathematical education, but their concern is not limited to related mathematics or to social mathematics. It has been rediscovered that there is a place in the high school curriculum for the traditional sequential courses in mathematics, a place of importance in the world of tomorrow.
The Case for Mathematics
Authors: Wickenden William E.
Published: January 2021

The long tide which swept our young people into high school and college over the last seventy-five years appears now to be mounting to its final surge. We are told that during these years our population increased three-fold and our high school enrollments ninety-fold, so that all but one-sixth of our young people reach high school and one-half complete its program. At the pre-war level, about one-sixth of all our young people entered college and about one-third of this number continued to their degrees. Inevitably, secondary education has been largely re-organized around social planning and personal adjustment as its coordinate foci, while the axis of higher education has shifted from the liberal to the functional area. How inevitable and how serviceable these changes have been, we who are close to the scene may find it hard to realize. A shrewd and sympathetic alien observer is likely to have them in truer perspective. For an appraisal of American secondary schools let me suggest a little book, The American Character written by a Cambridge University professor D. W. Brogan. My school-men friends call it the best thing of its kind and I am ready to agree.