bluuebunny/crossref_metadata_2025 huggingface dataset:
Membership Message
Authors: Rogers Mary C.
Published: November 2020

Another school year is well on its way. Vacation clays, with their many pleasant memories, have receded into the past—replaced by days and weeks of work and initial achievement. Plans for your new year are doubtless already being materialized and promise much in educational service through cooperative endeavor with those around you. National Council gives you its best wishes for the most successful and satisfying year in your professional experience.
Mathematics and Manpower
Authors: Orth Allen
Published: November 2020

Before accepting the invitation to discuss this subject, I debated with myself at some length as to my fitness to do an intelligent job. As an engineer by education, I do, or rather did, possess a certain familiarity with mathematics. I say “did” because in my present capacity I rarely come in touch with involved mathematics except on occasions like that of last month when I made out my income tax report.
A Report Card for Marking Achievement in Mathematics
Authors: Johnson Donovan A.
Published: November 2020

The task of assigning marks to students is a time consuming, reoccurring task of every teacher. Although it may be a tedious task, it is also one which affords the teacher an opportunity to guide, evaluate, and motivate the students and to establish contact with parents, employers and educational institutions. Despite these important functions of marks teachers are not satisfied with the adequacy or accuracy of our marking systems. It is the purpose of this paper to make specific suggestions for the improvement of marks in mathematics.
Notes on The History of Mathematics
Authors: Sanford Vera
Published: November 2020

The object of this article is to consider a quotation which, though not itself a problem, appears at the end of a set of miscellaneous examples in the Appendix to Roswell C. Smith's Practical and Mental Arithmetic.* The quotation concerns the presidential election of 1828.
Applications
Authors: Myers Sheldon S.
Published: November 2020

As important as general education may be it has always been my concern that, through over-emphasis on the general and the average, we may tend to neglect the gifted students who will provide the basis of our nation's technical supremacy in the future. Therefore, from time to time it may be appropriate in this department to include applications of a nature that will stimulate and challenge this type of student. Below are found two applications of one of the most fundamental patterns of mathematical reasoning, the principle of mathematical induction, which may profitably be brought to the attention of gifted juniors and seniors.1
Devices for A Mathematics Laboratory
Authors: Berger Emil J.
Published: November 2020

Anyone who has a learning aid which he would like to share with fellow teachers is invited to send this department a description and drawing for publication. Or if that seems too time-consuming, simply pack up the device and mail it. We will be glad to originate the necessary drawings and write an appropriate description. All devices submitted will be returned as soon as possible. Send all communications to Emil J. Berger, Monroe High School, St. Paul, Minnesota.
The President's Page
Authors: Mayor John R.
Published: November 2020

Two important meetings were sponsored by The National Council of Teachers of Mathematics in August. In addition to the importance of regular meetings added significance can be attached to these two conferences because each of them was different in several aspects from any previously sponsored by our organization.
Notes on The History of Mathematics
Authors: Sanford Vera
Published: November 2020

“After the Learner has been some time exercised in managing and transforming Equations, Order requires that he should try his Skill in bringing Questions to an Equation. And any Question being proposed, his Skill is particularly required to denote all its Conditions by so many Equations. To do which he must first consider whether the Propositions or Sentences in which it is expressed, be all of them fit to be denoted in Algebraick Terms, just as we express our Conceptions in Latin or Greek Characters. And if so, (as will happen in Questions conversant about Numbers or abstract Quantities) then let him give Names to both known and unknown Quantities, as far as occasion requires; and express the Sense of the Question in the Analytick Language, if I may so speak. And the Conditions thus translated to Algebraick Terms will give as many Equations as are necessary to solve it.
Devices for the Mathematics Classroom: Curves of constant breadth
Authors: Hazard William J.
Published: November 2020

There is exactly one curve having a constant. radius, the circle; but there is an infinite number of curves of constant breadth (or constant diameter). Some of these are occasionally used as cams.
Mathematical Miscellanea: An integraph
Authors: Kays George
Published: November 2020

Finding an area enclosed by an irregular boundary is a problem that is quite often difficult or tedious when solved by the routine formal methods of the integral calculus. The device discussed here is a simplification of a type of instrument used to solve problems in integral calculus. Figure 1 illustrates such a problem, that of finding the area enclosed by the x-axis, the lines x = a, x = b, and the curve of the (unknown) function y = F’(x).
Mathematics in the Junior High School: Textbook series and learning theory
Authors: Brown G. W.
Published: November 2020

Do textbook series in arithmetic develop the mathematical concepts and skills in accordance with current learning theory? This is an important question for the junior high school mathematics teacher. The answer determines the extent to which the teacher may rely on the textbook for guidance in the learning sequence. Since, for many students, mathematics instruction ceases at ninth grade, mathematical strands considered by teachers and textbook writers to be important must be brought to a culminating point by that time.
Some educational problems of significance to engineering colleges
Authors: Newsom Carroll V.
Published: November 2020

My responsibilities for higher and professional education in the state of New York include the maintenance of satisfactory standards of education for those students who desire to enter the seventeen professions recognized by the statutes of the state. This fact, combined with the experience of a quarter of a century of teaching college mathematics on both the undergraduate and graduate levels, including many classes that were composed chiefly of engineering students, has caused me to become genuinely concerned with many educational problems.
Devices for the Mathematics Classroom: A model for visualizing the formula for the area of a circle
Authors: Olander Clarence
Published: November 2020

Developing models and materials for visualization is important at all levels of instruction, but especially so at the junior high level.
Mathematics, the schools, and the ORACLE
Authors: Householder A. S.
Published: November 2020

It's true, we have machines to do our computing, but, the author asks, “Who is to understand what the machine is doing?”
Transforming the law of cosines for computational purposes
Authors: Greenberg Benjamin
Published: November 2020

The law of cosines has always frustrated those teachers who wish to use it computationally. The transformation developed in this paper makes it more amenable to the use of logarithmic computation.
Devices for The Mathematics Classroom: Casting geometric solids in plaster-of-Paris
Authors: Hainlin Wallace L.
Published: November 2020

Teachers of junior and senior high school mathematics occasionally experience difficulty in obtaining suitable models of common geometric solids for classroom use. Commercially made sets are ordinarily quite expensive, and many times it is impractical to have models made in the school shops. One fairly satisfactory solution to this problem is to cast such models in plaster-of-Paris by making use of paper models.
Devices for The Mathematics Classroom: A Model for Visualizing The Pythagorean Theorem
Authors: Olander Clarence
Published: November 2020

The Pythagorean relationship is one of the most widely used of all mathematical formulas. Accordingly, junior high school classes in algebra and general mathematics usually include some work involving simple applications of this important relation.
Historically Speaking,— The twelve base
Authors: Beard Ralph H.
Published: November 2020

For almost as many years as we have used our present system of number notation, it has been known that ten was not necessarily ten, but could represent two, or twelve, or any number that was used as the base of the number system. Ever since then, there has been continuing consideration of the possible advantages in substituting twelve for ten for the number base for general use.
Mathematics in the Junior High School: Remedial work in junior high school mathematics
Authors: Douglass Harl R.
Published: November 2020

Because of the very nature of mathematics learning the instructional materials in that field are necessarily sequential. While perhaps a greater degree of flexibility exists in seventh and eighth grades than in earlier grades, there is much material in seventh- and eighth-grade mathematics which can hardly be learned at all unless appropriate background concepts, principles, and skills in mathematics have been acquired in previous grades. In addition to these mathematical learnings there are also understanding, readiness, attitudes, and skills in problem solving, which are essential to success in the junior high school.
What is Going on in Your School?: A Comparison of Two Methods of Teaching a Unit on Insurance
Authors: Olander Clarence E.
Published: November 2020

Every teacher is interested in using a method of teaching that will provide for the students the most worth-while and life-like experiences which will lead to final achievement of the desired outcomes. Most of us have read a great deal of literature praising the merits of t he unit approach as a method of teaching. Some of us have become enthusiastic about the use of this method and probably have employed it successfully in our classes. Despite all the acclaim accorded the use of units of instruction, very little research has been conducted to find out how effective the unit approach has been in attaining our objectives. Thus, it would seem worth while to investigate the effectiveness of the unit approach in teaching a topic in mathematics.
What is Going on in Your School?: An Honors Program
Authors: Clark Norman
Published: November 2020

A vigorous academic challenge extending beyond the usual high school curriculum is the purpose of an လhonors programဝ at Lake Washington High School in Kirkland, Washington. The program, now on an experimental basis, takes students of exceptional academic promise through mathematics, logic, science, composition, history, literature, and humanistic courses that provide intellectual stimulation not common in the public schools. The student body and faculty have given the program whole-hearted approval and public reaction so far bas been highly favorable.
The meaning of mathematics
Authors: Springer C. E.
Published: November 2020

The spirit of mathematics is often violated by designating it as a “tool subject”thereby covering up a concept which pervades modern mathematics—the postulational method.
Mathematics in the Junior High School: Using mathematical recreations in the junior high school
Authors: Hall Arthur J.
Published: November 2020

What are mathematical recreations? What good are they? Where can they be found? How are they most effectively used? What research has been done on their use and effectiveness? These were the questions that an in-service seminar for mathematics teachers, under the direction of the writer, undertook to answer.
Mathematical Miscellanea: Two cube root curves
Authors: Sornito Juan E.
Published: November 2020

It is a matter of common knowledge that it is impossible to duplicate a cube by means of the straightedge and compasses. There are, however, several methods which make use of algebraic curves. This presentation is intended to suggest another such solution to the problem.
Tips for Beginners: Optional proofs of theorems in plane geometry
Authors: Lankford Francis G.
Published: November 2020

This department of The Mathematics Teacher for October contained an article on “Helping Pupils Use Proofs of Theorems in Geometry.” There, some suggestions were given for helping pupils to understand the nature of a deductive proof and to develop the ability to prove theorems independently.
What is Going on in Your School?: Geometry as an aid in teaching some facts about series discounts
Authors: Allegri Linda
Published: November 2020

In teaching series discounts one often has difficulty in convincing students that re-ordering the discounts does not affect the answer. By using area for two discounts and volume for three, one can give a concrete illustration of the commutability of the discounts in these two cases, and at the same time, obtain a formula for the single discount equivalent to the series.
An English schoolmaster looks at American mathematics teaching
Authors: Brace W. S.
Published: November 2020

An account of the insights into American education gained by one of our British “cousins,” while teaching in the Denver, Colorado schools.
Graphing in elementary algebra
Authors: Beberman Max , Meserve Bruce E.
Published: November 2020

Graphing in high school and college algebra has been too narrowly interpreted. This paper offers some good suggestions for broadening the topic to include other fundamental mathematical ideas.
Some comments on a simple puzzle
Authors: Rising Gerald R.
Published: November 2020

Puzzle situations stimulate interest if not used too frequently. This puzzle, while not new, may remind some teachers that mathematics can be taught through its use.
Pseudo logarithms
Authors: Larsen Harold D.
Published: November 2020

The drive to shorten the labor required to do tedious calculations is recorded by attempts to develop methods that did not stand the test of time.
Complex figures in geometry
Authors: Wiseman John D.
Published: November 2020

Here is more “food” for your gifted students.
Historically Speaking,— Living costs a century ago
Authors: Read Cecil B.
Published: November 2020

The author of a textbook often finds it difficult to keep current with prices, wages, and the like. Students are quick to pick up discrepancies between values stated in problems in a textbook and actual conditions in life. Nevertheless, although there may be some lag, textbooks do offer an interesting commentary on prices of approximately the period in which they are published.
Points and Viewpoints: Multitrack programs
Authors: Engen H. Van
Published: November 2020

A column of unofficial comment
Feed it back
Authors: Scheid Francis
Published: November 2020

Iterative methods are illustrated in such a simple way that high school students would enjoy getting acquainted with this important mathematical tool.
Sums of powers of integers
Authors: Yates Robert C.
Published: November 2020

Many advanced high school students will be interested in the formulas for integers, and the models may fascinate them.
Swords into plowshares
Authors: Turner Ethel M.
Published: November 2020

There are innumerable ways for a resourceful teacher to achieve his objectives.
Angle trisection—an example of “undepartmentalized” mathematics
Authors: Leo Brother
Published: November 2020

An old, old problem—but the emphasis is different, from the usual presentation.
One plus one equals toon
Authors: McFeely Franklin S.
Published: November 2020

Names for binary numbers are proposed. The names are similar to those for decimal numbers with the same representation.
Sets, sinners, and salvation
Authors: Meder Albert E.
Published: November 2020

The sinner (mathematics teacher) is tempted. He thinks, “to do something really worthwhile is time-consuming, laborious, and difficult. Therefore, let us teach sets.’
A brief history of the dollar mark
Authors: Murray Sharon Ann
Published: November 2020

Contrary to widespread belief, our dollar mark did not originate in the combination of the letters U and S. The symbol actually has a Spanish origin.
Primitive Pythagorean triples
Authors: Moshan Ben
Published: November 2020

An ancient problem continues to fascinate and challenge. High school students know about 3, 4, 5 right triangles. Do they know about 8, 15, 17 right triangles and 861, 620, 1061 right triangles?
The trapezoid—and area
Authors: Hinkel Wilfred H.
Published: November 2020

An informal approach to the formulas for the areas of the usual polygons studied in the junior high school.
A further note on dissecting a square into an equilateral triangle
Authors: Hawley Chester W.
Published: November 2020

The dissection of a square into an equilateral triangle requiring only three cuts, with no part turned over.
New Ideas for the Classroom: Concerning simultaneous solutions of polar equations by the graphical method
Authors: Hoffman A. A. J. , Osborn Roger
Published: November 2020

Current proposals for new topics for the mathematics class suggest topics such as inequalities and polar equations. The articles below briefly describe ways of making these topics more meaningful. You can stimulate interest and help teachers introduce new topics if you will send your successful experiences to the editor.
Going somewhere?
Authors: Ore Oystein
Published: November 2020

What's the shortest route from here to there?
The law of contraposition
Authors: Tierney John A.
Published: November 2020

Some examples are given to show how students can make use of the law of contraposition in the solution of some problems.
Minimum multiplying magic squares
Authors: Gilbert Jack
Published: November 2020

Here are magic squares constructed on products rather than on sums.
The binomial theorem
Authors: Stewart Lurline
Published: November 2020

An example shows how a teacher may enrich the students’ learning of the binomial theorem.
New Ideas for the Classroom: An intramural mathematics league
Authors: Westby Harold
Published: November 2020

If you are looking for a new way to promote a mathematics club or, more important, increased achievement in mathematics, try an intramural league with homeroom mathematics teams competing against each other. An intramural mathematics league has been highly successful at my school.
Applications of finite arithmetic, II
Authors: Dubisch Roy
Published: November 2020

An application of finite arithmetic to the problem of coding and uncoding messages is presented here.
The SMSG geometry program
Authors: Moise Edwin E.
Published: November 2020

What should be done about geometry leaching in the high school program? The School Mathematics Study Group has some promising answers.
The role of the state supervisor of mathematics
Authors: Hawthorne Frank S.
Published: November 2020

What does a state supervisor of mathematics do?
Calculus in the high school
Authors: Ferguson W. Eugene
Published: November 2020

Calculus is a proper high school subject, if taught by qualified teachers to adequately prepared students.
Historically Speaking,—: Abū al-Wafā' on the solar altitude
Authors: Nadir Nadi
Published: November 2020

This paper deals with a fundamental problem of ancient astronomy, that of determining the time of day by observing the solar altitude. It is based on a short treatise by Abū al-Wafā' al-Būzjānī [l]* (fl. 970 a.d.), who is best known as having been one of the discoverers of the sine theorem in spherical trigonometry. In his introduction Abū al-Wafā' states that there has been talk to the effect that a certain rule stemming from Habash alHasib (fl. 850) is only approximate and not susceptible of proof. Abū al-Wafā' then proceeds to give three proofs of the rule. What is of more interest is that the same rule is given in a work [2] of the Hindu mathematician Brahmagupta (fl. 650). We are thus presented, not only with an incident in the emergence of trigonometry from spherical astronomy, but also with a speci fic example of how Hindu science penetrated that of the Islamic world.
Tips for Beginners: Using number theory to foster interest in general mathematics
Authors: Fitzgerald William M.
Published: November 2020

The theory of numbers offers a great deal of potential for enriching our mathematics curriculum at all levels. Some of these topics can be used to inspire the be-low-average students as well as the average and bright students. An interesting example is reported by Goodman,* who used children's blocks to demonstrate the basic ideas of factors and primes.
The transition from school to college mathematics
Authors: Ogilvy C. Stanley
Published: November 2020

“How can I inject more inspiration and interest into the topics already at hand?”
Remarks on the teaching of calculus in the secondary school
Authors: Blank Albert A.
Published: November 2020

What mathematics should he offered to students in the twelfth grade?
Some geometric ideas for junior high school
Authors: Brune Irvin H.
Published: November 2020

“The race, including those who advocate abstractions today, learned much of its mathematics through practical needs and by the way of real problems…. Surely a small share of such learning while solving, abstracting while applying, and generalizing while testing should not be denied young people today.”
Divisibility by two
Authors: Parsons Kenneth B. , Franklin Stanley P.
Published: November 2020

An interesting theorem relating properties of odd and even integers to number notation in bases other than ten.
Discovery
Authors: Willoughby Stephen S.
Published: November 2020

The discovery method may be more effective when the teacher does not know precisely where he is going and how to get there.
Experimental Programs: Concepts of mathematics—a unique program of high school mathematics for the gifted student
Authors: Rollins Wilma E. , Brown Stanley P. , Johnson Arnold O. , Soychak Anthony G. , Dionne Peter L. , Bennet Albert B.
Published: November 2020

For the past two years, the Maine State Department of Education under a Title VII grant from the National Defense Education Act has sponsored the television program, “Concepts of Mathematics.” Three commercial television stations in Maine and one educational station in New Hampshire have cooperated in televising this program of advanced high school mathematics to capable high school sophomores, juniors, seniors, and adult viewers. The objective of this program was to identify and evaluate an economical and practical method of providing intellectual stimulation to gifted high school students.
Tips For Beginners: Altitudes, medians, angle bisectors, and perpendicular bisectors of the sides of triangles
Authors: Schor Harry
Published: November 2020

Students have some difficulty with the concepts and constructions of the altitudes, the perpendicular bisectors of the sides, the medians, and the angle bisectors of the triangle. Some students seem to show little understanding of the words of the definitions, and the mechanics of straightedge and compass construction often do not help to clarify their understanding of the definitions.
Implementing a new mathematics curriculum
Authors: Henkelman James
Published: November 2020

An overview of past and present efforts to improve mathematics teaching and the in-service training of teachers
Mathematics teachers and self-evaluation procedures
Authors: Simpson Ray H.
Published: November 2020

“He hasn't had twenty years’ experience. He has had one year's experience twenty times.”
Historically Speaking,—: The development of modern statistics
Authors: Varberg Dale E.
Published: November 2020

That area of study which we now call statistics has only recently come of age. While its origins may be traced back to the eighteenth century, or perhaps earlier, the first really significant developments in the theory of statistics did not occur until the late nineteenth and early twentieth centuries, and it is only during the last thirty years or so that it has reached a full measure of respectability.
Using vectors to solve simultaneous equations
Authors: Salzarulo Leonard M.
Published: November 2020

Relating algebra to trigonometry
The case against calculus
Authors: Allendoerfer Carl B.
Published: November 2020

Calculus is frequently taught at the wrong time, by the wrong people, and in the wrong way. It is high time we gave this matter our urgent attention.
A variation on binary notation
Authors: Kingston J. Maurice
Published: November 2020

An idea for the student who wants to explore
Some thoughts on quotient and remainder patterns
Authors: Moore Alan S.
Published: November 2020

Every teacher of mathematics is familiar with the principle of “casting out nines.” The principle is based on the fact that when any number is divided by nine, the remainder obtained is the same as that obtained when the sum of the digits of the number is divided by nine. For example, on dividing 75,808 by nine, we obtain a quotient of 8,429 with a remainder of 7, the latter number being the same as the remainder obtained when the sum of the digits of the dividend, namely 34, is divided by nine.
On the learning of mathematics by children
Authors: Fitzgerald William M.
Published: November 2020

Teaching should increase the diversity of the achievements of students rather than the uniformity.
Strategies of proof
Authors: Robinson Edith
Published: November 2020

A look at the common strategies of proof used in high school mathematics
Another format for proofs in high school geometry
Authors: Tenney Arthur E.
Published: November 2020

A style of writing proofs that uses the usual outlining procedures
Structure diagrams for geometry proofs
Authors: Thorsen Carolyn C.
Published: November 2020

A dynamic way to present geometry proofs
Learning and proficiency in mathematics
Authors: Gagné Robert M.
Published: November 2020

An account of a method and some recent findings in experimental studies of learning mathematics
Historically Speaking,—: Johan de Witt's kinematical constructions of the conics
Authors: Easton Joy B.
Published: November 2020

In second-year algebra courses the conic sections are defined as type-quadratic equations and the graphs of the curves plotted. It is difficult at this level to relate these equations to the usual geometric definitions of the curves. There arc, however, a number of little-known locus definitions which lead immediately to the canonical equations through the application of elementary theorems of proportion which should be familiar to students of plane geometry.
Tips for Beginners: What to do in a mathematics club
Authors: Granito Dolores
Published: November 2020

While a few articles have appeared, describing the goals and organization of mathematics clubs, very little concern has been paid in those articles to the specific activities that make such clubs dynamic and interesting to sponsoring teachers and student members alike.
A queueing simulation
Authors: Gramann Richard H.
Published: November 2020

An article that will give the reader something to think about the next time he has to stand in a line
Historically Speaking,—: How modern is modern mathematics?
Authors: Schaaf William L.
Published: November 2020

We hear a great deal these days about “modern mathematics” in the schools or, perhaps just as frequently, the “new” mathematics. These terms are used interchangeably. The term “modern mathematics” has suddenly become a byword among teachers and educators, not without evoking some controversy.
A list of standard corrections
Authors: Wernick William
Published: November 2020

A teacher of geometry often finds himself writing the same corrections and criticisms over and over again on students’ papers, because, sadly, the same mistakes occur over and over again.
Designing RAPID, an analogue computer
Authors: Slook Thomas H.
Published: November 2020

How mathematical concepts are used in analogue computer design
The Kensington stone mystery
Authors: Struik D. J.
Published: November 2020

One day in 1898, we are told, the Swedish immigrant farmer Olof Ohman, clearing a timbered section of his land near Kensington, Minnesota, found a stone under the roots of a tree.
Structuring a Proof
Authors: Klingler Donn L.
Published: November 2020

A method of writing that displays the structure of a proof
Two Decades of Mathematical Education in Puerto Rico
Authors: García Mariano
Published: November 2020

Mathematics curriculum reform in Puerto Rico
An Example of Student-Generated Sequences in Mathematics Instruction
Authors: Kaplan Jerome D.
Published: November 2020

In what sequence do students ask questions when they want to learn?
Tips for Beginners: A New Look at Enrichment
Authors: Antone Malerich
Published: November 2020

Judging by the number of projects which occupy our youngsters—the posters, the scrapbooks, the collections of various kinds, the excellent work entered at science and mathematics fairs, the Math Weeks, and the student-prepared assembly programs—it would seem that the idea of enrichment in a true sense of the term has “taken hold” in elementary and secondary mathematics teaching.
Discovery of Patterns in the Difference of Two Squares
Authors: Smith Joe K.
Published: November 2020

Many mathematical discoveries are made by the really talented mathematics students. They enjoy making discoveries, and it is a pleasure to help them. No one is especially surprised when the really bright students make discoveries. But sometimes students with limited background and ability in mathematics make discoveries which are interesting and exciting. Evidence that students of limited ability can make exciting discoveries came to the writer recently while he was teaching a noncredit, remedial arithmetic class at the college freshman level.
Membership Report
Authors: Gates James D.
Published: November 2020

Our rate of membership growth has continued to be large, as indicated by the totals in the report which follows. We can all be proud of the increase of 4,103 individual members and 4,311 institutional subscriptions.
Historically Speaking,—: The volume of a Truncated Pyramid in Ancient Egyptian Papyri
Authors: Gillings R. J.
Published: November 2020

The Moscow Mathematical Papyrus, which was acquired by the Moscow Museum of Fine Arts from the Egyptologist Golenischeff in 1912, contains 25 problems of varying interest and importance. Problem No. 14 shows quite clearly that the Egyptian scribes were familiar with the formula.
Geometric Diversions: a 25-Point Geometry
Authors: Coxford Arthur F.
Published: November 2020

How many times in teaching geometry have you come to the days before Christmas or Easter vacation knowing that very little would be accomplished in that time? And yet, not wishing to waste the time entirely, you sought something that would be worthwhile, but also different, in order to capture the fancy of your students.
Dabar, logos, and shhhh
Authors: Perry Marcia
Published: November 2020

One of the big problems in mathematics education today centers around the importance of words in our thinking and understanding. Do they help or hinder, stimulate or stifle? In the midst of such wonderings, it is interesting to note how the concept of “word” has changed through the ages.
Points And Viewpoints: Authors please note
Authors: Brune Irvin H.
Published: November 2020

A column of unofficial comment
Introducing number theory in high school algebra and geometry, Part 2, Geometry
Authors: Barnett I. A.
Published: November 2020

A new look at some classical ideas
Experimental Programs: Educational research and the mathematics educator
Authors: Scandura Joseph M.
Published: November 2020

The purpose of this paper is not to review all or even much of the past or current research in mathematics education; this has been done elsewhere. Monographs covering research in mathematics education up to about 1960, for example, are available from the U.S. Department of Health, Education and Welfare. Some of the NCTM yearbooks also provide a good source.
Historically Speaking,—: The world of polyhedra
Authors: Wenninger Reverend Magnus
Published: November 2020

Geometrical solids can become the subject of a fascinating study. Not everyone, of course, will want to make the attempt to understand all the theoretical mathematics involved in discovering and classifying those solids known as uniform polyhedra and their stellated forms. But everyone surely can appreciate the beauty and symmetry of these solids, whose history is as ancient as Plato, Archimedes, and Euclid and as modern as Coxeter, today a well-known professor of mathematics at the University of Toronto.
Tips for Beginners: A compass-ruler method for constructing ellipses on graph paper
Authors: Kaner Samuel
Published: November 2020

One of the well-known techniques for constructing an accurate ellipse consists of drawing two concentric circles from the center of the ellipse, having one radius equal to the semiminor axis and the second radius equal to the semimajor axis.1
Deductive proof of compass-ruler method for constructing ellipses
Authors: Snyder Henry D.
Published: November 2020

One of the most widely accepted objectives of the current approaches to teaching mathematics is to provide a meaningful blend of intuitive understanding and deductive verification. Mr. Kaner's article provides a novel situation in which both aspects are present.
Using high school algebra and geometry in Doppler satellite tracking
Authors: Thompson Robert A.
Published: November 2020

A chance for students to apply algebra and geometry to a space-age problem
The relation between distance and sight area
Authors: Chi-Ming Chow
Published: November 2020

We all know that the apparent size of a moving body seems smaller to us as it moves farther away from us. According to this phenomenon, let us define the apparent length of a body as “sight length” and the distance between our eye and the body as “observing distance.” Thus, we can develop the following theorem.
Conditions governing numerical equality of perimeter, area, and volume
Authors: Smith Leander W.
Published: November 2020

Development of theorems leading to pattern generalization
Divisibility patterns in number bases
Authors: Stastny Sister M. Barbara
Published: November 2020

The reason for studying various number bases is the better understanding of principles.
A general test for divisiblity by any prime (except 2 and 5)
Authors: Bold Benjamin
Published: November 2020

Is there a simple test for divisibility by any prime?
Fermat's last theorem
Authors: Duncan Hilda F.
Published: November 2020

A new look at a genuine problem
International Mathematical Education: Methods and techniques of explaining new mathematical concepts in the lower forms of secondary schools, Part 1
Authors: Papy G.
Published: November 2020

Any child who is at present in the 12 to 15 age group is likely to have to use mathematics later, whatever his occupation, as a means of understanding, enquiry, and problem-solving.
What about modern programs in mathematics?
Authors: Payne Holland
Published: November 2020

A look at the utility of modern materials
A note on measurement
Authors: Hannon Herbert
Published: November 2020

Toward more convenient notation
Legibility of mathematical tables: the Babbage experiments of 1827
Authors: Sueltz Ben A.
Published: November 2020

In 1827, Charles Babbage published his Table of Logarithms in London. In the preparation of the first edition, the author, together with the printer, carried out one of the first systematic investigations into the factors which make printed pages both easily legible and optically endurable. In the preface, Babbage summarizes the results of his studies and experiments with sizes and arrangements of type as well as with the color and texture of paper. That the Babbage experiments and conclusions are valid is readily apparent when one compares his book with similar volumes printed more than one hundred years later.
Precision, accuracy, and other ambiguities
Authors: Thomas Gerald R.
Published: November 2020

One of the goals of secondary school physical science courses should be to make students aware of the fact that the numbers they work with in science are approximations, not exact values. Methods of working with approximate data can conveniently be presented in science courses because of the intrinsic need for them. Mathematics classes can contribute some background concepts while discussing various topics.
The ubiquitous number five
Authors: Barnett I. A.
Published: November 2020

Five is not merely half of our base.
Historically Speaking,—: Adam Riese
Authors: Carpenter Dorothy I.
Published: November 2020

Reflect, for a moment, on the reeducation and records-changing entailed if we were to replace our illogical weights-and-measures systems by systems employing metric units. Such a Gargantuan task probably dooms us to the status quo. Yet a comparable change was effected in Europe during the period from the twelfth century to the sixteenth century as people gradually rejected Roman numerals in favor of the Hindu-Arabic. Many advocates of the new system have long been forgotten by even their own countrymen, but not Adam Riese. It was this arithmetic teacher and textbook writer who was largely responsible for the German acceptance of computation with numerals in lieu of the counting board. So influential was he that now, more than four centuries later, his name is still being perpetuated in the folk phrase nach Adam Riese used in connection with results in arithmetic computation, much as we might say that something is “according to Hoyle.”
Historically Speaking,—: Mathematics in ancient Egypt: a checklist (1930-65)
Authors: Guggenbuhl Laura
Published: November 2020

In 1929, Chace, Bull, and Manning published the second volume of a two-volume definitive work on the Rhind Mathematical Papyrus. This publication has become an undisputed classic, and one wonders why it has not appeared as a paperback reprint.
Retention as a function of paired and individual use of programmed instruction
Authors: Dick Walter
Published: November 2020

One of the current issues in the field of programmed instruction is the effect of response mode upon learning and retention. A number of studies have indicated no difference in immediate learning as a function of various forms of overt, covert, and reading responses. However, Krumboltz and Weisman [1]* found that their written response group, when compared to those who just thought their responses and those who read the correct responses, showed significantly greater retention after a delay of two weeks. Goldbeck and Campbell [2] had eighth graders study a 32-frame program on electricity in order to test for differences in performance as a function of learning response; they used reading, writing, and thinking response groups. The results indicated significantly greater retention for the reading group after a period of ten weeks. Tobias and Weiner [3], using essentially the same design as Goldbeck and Campbell, found no differences among their groups on either immediate or delayed recall. The programs used in these studies varied in length from 32 to 177 frames.
Divisibility tests of the noncongruence type
Authors: Jordan John Q.
Published: November 2020

Does your supply include tests for seven and thirteen?
Experimental Programs: Textbook difficulty and mathematics achievement in junior high school
Authors: Nelson L. Doyal
Published: November 2020

The importance of the role of the text-book in determining the content, organization, and mode of presentation of school mathematics can hardly be overemphasized. In recent years various groups and individuals have devoted considerable time and effort to the production of school mathematics textbooks with content strikingly different from that found in traditional textbooks. Probably more striking are the changes which have occurred in the methods of organizing and presenting the material. These changes have generated increased interest in the question of whether there is any one method of presenting mathematical ideas which is superior to all others in promoting maximum learning efficiency on the part of students.
Geometric Techniques for Graphing
Authors: Haddock Glen , Hight Donald W.
Published: November 2020

Mathematics, both pure and applied, is primarily concerned with functions. These range from a simple constant function to complex functional relationships used to determine paths of rockets. Practicing scientists in all disciplines are confronted with the need to readily formulate, analyze, and apply functions.
Are Circles Similar?
Authors: Johnson Paul B.
Published: November 2020

While similarity is one of the major topics of secondary geometry, most courses consider only similarity of triangles and polygons.
It's Time to Take a Closer Look at High School Calculus
Authors: Beninati Albert
Published: November 2020

We are just emerging from a decade of unprecedented activity in mathematics education. It may well be a good time to take a closer look at one aspect of this activity that has developed in the past few years—one that could be enmeshing us in a rather ugly situation.
Points and Viewpoints: Welcome to a New Department
Authors: Brune Irvin H.
Published: November 2020

“Classics in Mathematics Education” comes to light in this issue. It fits in with a noteworthy sequence of excellent departments that have enhanced the MATHEMATICS TEACHER through the years. We are most happy to welcome it and wish it a long life.
Classics in Mathematics Education: The Human Worth of Rigorous Thinking
Authors: Keyser Cassius J.
Published: November 2020

The question I purpose to discuss briefly is this: How much mathematical discipline is essential to the appropriate education of men and women as human beings?
Axiom or Theorem?
Authors: Leake Lowell
Published: November 2020

The statement often appears in mathematical literature that the axioms chosen in any mathematical system are relatively arbitrary. Sometimes it is stated later that a different choice of axioms would lead to the same theorems. Occasionally, various writers indicate that when two different sets of axioms for the same mathematical system are available, choosing one set allows the mathematician to prove the other set as theorems and vice versa. The student of mathematics meets such statements many times in his studies, and he is also told that a major characteristic of mathematics is its axiomatic and deductive nature. Indeed, one of the features of the current revolution in the teaching of mathematics is the emphasis it gives to the axiomatic structure of all branches of mathematics. The alert and wise teacher, of course, points out these characteristics at appropriate moments in the classroom.
A Teacher-Made Slide Program
Authors: Xavier Sister Ann
Published: November 2020

Anyone interested in the teaching of secondary school mathematics cannot but be impressed and overwhelmed by the vast number of texts and visual aids which have been produced within the past few years. It came, therefore, as a surprise to find, when looking for a filmstrip on the introduction of the trigonometric functions, that none was available. Visual aids on the subject of trigonometry seemed limited to numerical applications in the right triangle. Out of this fruitless search was born a challenging and satisfying project. If we could not purchase the filmstrip we wanted, we would produce our own.
On the Evaluation of ∫ sec x dx and ∫ csc x dx
Authors: Hellman M. J.
Published: November 2020

When the integration of sec x and csc x is first attempted in the elementary calculus course it becomes clear that these quantities are not the end products of differentiating any simple functions the student has yet encountered. It appears to the author that the best technique and one which is most naturally motivated is to examine the simple trigonometric identities already encountered to see whether any can be found which relate sec x and csc x to functions which are more readily integrated, that is to functions which are recognizable as end products of differentiating elementary functions.
Fields and Near-Fields of Ordered Pairs of Reals
Authors: Dorman Janet , Tollefson Jeffrey L. , Stein F. Max
Published: November 2020

A Complex number may be written as a + ib or as an ordered pair of numbers (a, b), with a and b real. Also a rational number may be written as c/d or as the ordered pair (c, d), d≠0, with c and d integers. Then in each case, after a definition of equality is made, the binary operations of addition and multiplication can be considered.
The Teaching of Remedial Mathematics
Authors: Meserve Bruce E.
Published: November 2020

As a nation we pride ourselves that half our our high school graduates enter college. Yet many of these people will not graduate. Moreover, five years from now nearly 70 percent of all jobs will probably require at least two years of college education. We note that in the early 1950's about 11 percent of all college students were in junior colleges. Now about one million students are enrolled in two-year institutions; this represents about 20 percent of all students in college and about twice the number in teachers’ colleges. With the help of local, state, and federal funds junior colleges are growing very rapidly. It is clear that junior colleges play a vitally important role in our program of higher education.
Programming and Automation
Authors: May Kenneth O.
Published: November 2020

The “programmed learning” movement traces its ancestry to Sydney L. Pressey, who in 1926 observed that his multiple-choice testing machine also performed a teaching function. The current movement was launched by B. F. Skinner and associates, who in the 1950's devised teaching machines inspired by experiments on animals.
Mathematics Teachers, On Guard!
Authors: Meserve Bruce E.
Published: November 2020

It has become an NCTM tradition to have a presidential “state of the council” address immediately preceding our annual business meeting. Last year I described in detail our very healthy, professionally active, financially sound organization. This year the title that I have selected is purposely ambiguous. I hope that each of you will find in my remarks several challenges, several alerts for your daily professional activities, and the recognition of several situations that you will want to encounter with great care. Mathematics teachers, on guard! Alert! Watch out!
Historically Speaking—: Radii of the Apollonius Contact Circles
Authors: Mills C. N. , Eves Howard
Published: November 2020

The subject of this article is not new. In the October 1961 issue of The Mathematics Teacher, Professor N. A. Court gives an excellent history of the Apollonius contact circles. Before the invention of analytic geometry, mathematicians were concerned with the construction of the circles. Descartes attempted an analytical solution, and we learn from his memoirs that he gave up in despair.
What is a Trapezoid?
Authors: Keedy M. L.
Published: November 2020

Terminology in the “new” mathematics is different from the so-called traditional, but it is certainly not consistent within the new literature. New terms have been invented and old terms redefined in an attempt to make terminology more consistent for a given book or series of books. Each author is, of course, entitled to define his mathematical terms as he sees fit, but it would be helpful if “modern” terminology could be more or less standardized. To this end, it is proposed that there be some public debate relative to the merits of certain definitions. Once the reasons for and against using a particular definition are clarified, it will be possible to make a choice that is wiser than those choices dictated largely by personal taste, bias, or habit. In this note I speak as a proponent for a nontraditional definition of trapezoid, hoping that those who disagree will make known their views as well as their reasons for disagreeing.