bluuebunny/crossref_metadata_2025 huggingface dataset:
What is Going on in Your School?: Mathematics and guidance in Kansas schools at the ninth-grade level
Authors: Elliott Earl S.
Published: November 2020

Many school administrators recognize the importance and value of guidance services to the individual. However, relatively few pupils to date have had the opportunity to reap the values of a guidance program.
A note on “variable”
Authors: Bennett Albert A.
Published: November 2020

It is important that teachers give serious thought to the meaning of the basic terms used in their classes. This note will help to clarify the term “variable.”
The place of the variable in the teaching of mathematics
Authors: Rich Barnett
Published: November 2020

The general concept of a variable is a difficult one. This paper offers suggestions for highlighting some important aspects of this concept.
General mathematics for terminal students in California junior colleges
Authors: Rowe Jack L.
Published: November 2020

A report on the efforts of one junior college to improve the mathematics curriculum. How to stimulate an interest in mathematics at this level needs further study.
Symbolized theorems
Authors: Clough Meriam A.
Published: November 2020

An interesting device for symbolizing theorems is presented here. It should not, however, lead to overemphasis on writing detailed proofs throughout the year of geometry in the secondary schools.
A heart for Valentine's Day
Authors: Keiber Mae Howell
Published: November 2020

What could be more appropriate for the mathematics class than a heart construction for Valentine's Day—especially if the construction is geometric? The valentine heart is a cardioid, but its construction can be simplified to involve the constructions of plane geometry.
The concrete basis of the abstract
Authors: Sawyer W. W.
Published: November 2020

The Commission on Mathematics recommends that the postulates of a field should form the basis for organizing a course in first-year algebra. This article raises the general question of how axioms come into algebra.
Current school mathematics curricula in the Soviet Union and other Communist countries
Authors: Wirszup Izaak
Published: November 2020

Russians follow developments in American education with interest. Most certainly we should acquaint ourselves with the basic Russian curricular pattern.
Trisecting an angle
Authors: Robusto C. Carl
Published: November 2020

In view of the interest many of the “uninitiated” show in angle trisection, commentary on a proof that trisection is not possible, and presentation of the proof of impossibility, may serve a purpose.
Officers of the NCTM Affiliated Groups
Authors: Roudebush Elizabeth
Published: November 2020

The National Council of Teachers of Mathematics as a dynamic, growing organization furnishes leadership in mathematics education in the United States of America and Canada. At present the membership of more than 20,000 comes from all parts of the world. The publications of NCTM are excellent, and give help and inspiration to every member.
Logarithmic and exponential functions —a direct approach
Authors: Seebeck C. L. , Hummel P. M.
Published: November 2020

A direct approach to logarithms rather than the indirect approach as an inverse of an exponential function has many advantages for teaching secondary school students.
Mathematics program outline for junior and senior high schools
Authors: Keaveny William P.
Published: November 2020

To help pupils from a wide range of mathematical backgrounds and. abilities learn, mathematics commensurate with their potential, a wide variety of courses must be offered.
How to use the perpetual calendar
Authors: Kessler Donald Joe
Published: November 2020

Why does it work?
Preparation in mathematics of mathematics teachers
Authors: Kinsella John J.
Published: November 2020

What should the mathematics teacher's preparation be for the years after 1963?
Annual financial report
Authors: Ahrendt M. H.
Published: November 2020

The report below gives a brief picture of the financial condition of the Council as of May 31, 1960. Readers who wish to compare this report with the report for the previous fiscal year will be able to make some interesting comparisons.
Why arithmetic works
Authors: Levi Howard
Published: November 2020

Most of us agree that it is more efficient to enlist the pupil's mind rather than to impose a mechanical routine on him.
Toward autotelic learning of mathematical logic
Authors: Allen Layman E.
Published: November 2020

Students may have enjoyment playing games and learn some mathematical logic at the same time.
A device for demonstrating some elementary properties of integers
Authors: Cohn Ernst M.
Published: November 2020

Simple, effective models for introducing students to elementary number theory
The geometric progression presented geometrically
Authors: Noya Louis E. De
Published: November 2020

A geometric analysis of the sum of an infinite geometric series
Pattern multiplication
Authors: Page Robert L.
Published: November 2020

Multiplication of whole numbers is often defined in terms of addition. As numbers increase in size, however, the process of “additive multiplication” grows correspondingly cumbersome. The ordinary method of multiplication can likewise he tiresome for large numbers. Because of their symmetry, certain numbers may be easily and rapidly multiplied by “pattern multiplication.”
The case of the two Sir William Hamiltons
Authors: Eves Howard
Published: November 2020

There is probably no greater source of confusion in the history of mathematics (indeed, in the history of any subject) than the perpetuation of an error of fact unknowingly made by some authority in the field. Such errors are innocently carried on and multiplied by subsequent writers leaning on either the original authority or on someone who did lean on the original authority.
The role of physics in the teaching of mathematics
Authors: Fehr Howard
Published: November 2020

Teachers of mathematics are by definition responsible for teaching mathematics. Good mathematics teachers are also interested in the general education of their students.
Tips For Beginners: Unilateral mathematics teaching
Authors: Kovach Ladis D.
Published: November 2020

One of the basic concepts in mathematics is the concept of inverse operations. Any mathematical operation worth its salt has an inverse—one that “undoes” the operation. The inverse of addition is subtraction, of multiplication is division, of involution is root extraction, of differentiation is integration, etc. Teaching these without mentioning the associated inverse operations can be called unilateral teaching.
Reviews of films
Authors: _ _
Published: November 2020

Some films prepared for use in secondary school mathematics classes and in teacher education classes are listed and reviewed.
Tips for Beginners: Teaching inductive proofs indirectly
Authors: Shreve Warren E.
Published: November 2020

In a presentation of mathematical induction many high school students do not grasp the inductive concept readily. Further, aligned dominoes or toy soldiers do not always put the idea of mathematical induction within their reach. Presented here are two examples which give different methods of making an inductive proof, one directly and the other indirectly.
Experimental Programs: An experiment in the teaching of college algebra
Authors: Dobyns Roy A.
Published: November 2020

In recent years there has been tremendous activity and interest in programmed instruction in academic circles and among psychologists.
Permutations from a different point of view
Authors: Comfort Joseph J.
Published: November 2020

An introduction to the study of groups
Irrationals, Area, and Probability
Authors: Myers Donald E.
Published: November 2020

An interpretation of the area of a set D as the probability that a point chosen at random from R will be in D
Tips for Beginners: A Trigonometric Recapitulation
Authors: Wallender Arnold L.
Published: November 2020

Regardless of how diligently one stresses the structure and fundamental concepts of mathematics in a course at the elementary level, it is easy to fade into the manipulative and computational aspects of so called practical problems. This is particularly so in a trigonometry course or in the units of an analysis course that deal with trigonometric functions.
Historically Speaking,—: Saccheri, Forerunner of non-Euclidean Geometry
Authors: _ _
Published: November 2020

Few people contest the statement that Lenin, the father of the Russian Revolution, effected the most drastic social upheaval of our modern times. Nor do they argue whether or not the monuments erected or the volumes published to commemorate this feat are in proportion to the enormity of the achievement.
Tips for Beginners: Discovering the Centroid of a Quadrilateral by Construction
Authors: Kaner Samuel
Published: November 2020

Several simple experiments concerning lever problems have been introduced in current mathematical literature. In Mathematical Snapshots, by H. Steinhaus, one finds the center of mass of a uniform rod by supporting the rod with each hand and then carefully moving the hands together until they meet at the point below the center of balance. SMSG materials for the junior high school also teach simple laws of levers by experimentation and generalization.
Leave the Door Open!
Authors: Adkins Julia
Published: November 2020

One summer we had a demonstration class, using students who had just completed the tenth grade.
What's New in Teaching Slow Learners in Junior High School?
Authors: Greenholz Sarah
Published: November 2020

Helpful hints directly from the firing line
A Demonstration of Indeterminate Forms Using Finite Methods
Authors: Williams Horace E.
Published: November 2020

The teaching of traditional units in this area of calculus may be facilitated by the use of numerical demonstrations.
Spinning Out the Square Root of an Integer
Authors: Draim N. A.
Published: November 2020

A novel algorism for an exercise that has interested people for centuries
A coordinate approach to the 25-point miniature geometry
Authors: Heidlage Martha
Published: November 2020

What if we limit the number of points and lines?
The history of the dodecahedron
Authors: Phillips J. P.
Published: November 2020

Of the five regular solids, the dodecahedron has many claims to the most complex history. Unlike the others (cube, tetrahedron, octahedron, and icosahedron), which originated before historical records were kept, tradition assigns the discovery of the dodecahedron to Hippasus, a Pythagorean of the fifth century before Christ [1].* For his impiety in devising an addition to the perfect solids given by the gods he was lost at sea, according to the legend.
The use of transformations in deriving equations of common geometric figures
Authors: Perisho Clarence R.
Published: November 2020

A clear treatment of useful geometric transformations
Checking computations in nondecimal bases
Authors: Rosenberg Edwin A.
Published: November 2020

Theoretical aspects of a question pupils sometimes raise
Use of discriminant analysis for selecting students for ninth-grade algebra or general mathematics
Authors: Ivanoff John M. , De Wane Evermode T. , Praem O.
Published: November 2020

Helping pupils to make better choices in courses in mathematics
A radical approach to ab=ab
Authors: Pedley Arthur H.
Published: November 2020

Definitions are the heart of the matter.
Experimental Programs: An interim report on the national longitudinal study of mathematical abilities
Authors: Cahen Leonard S.
Published: November 2020

In September of 1962 approximately 110,000 students took their first set of inventories as participants in the National Longitudinal Study of Mathematical Abilites (NLSMA).1
Experimental Programs: A study of the factors associated with success in first-year college mathematics
Authors: Wick Marshall E.
Published: November 2020

The past few years have witnessed what has been described by many as a revolution in school mathematics. Although this revolution is little more than ten years old, and has attained its full momentum within the past five years, its effects have been felt all the way from the elementary school to the college undergraduate program. The mathematics curriculum in grades seven through twelve has been the area of greatest activity, with the college-preparatory program receiving the greatest attention. The variety of new materials, textbooks, and entire new mathematics programs that has been developed for these grades is well known to those interested in the teaching of mathematics.
The Circle of Unit Diameter
Authors: Garfunkel J. , Leeds B.
Published: November 2020

Using trigonometric relations to establish geometric theorems is not an unfamiliar procedure, but the introduction of a circle of unit diameter acts like a catalytic agent and facilitates such proofs greatly.
Experimental Programs: Reading and Mathematics
Authors: Call Russell J. , Wiggin Neal A.
Published: November 2020

The teaching of the general topic of solving word problems in mathematics is often a topic of discussion at meetings of mathematics teachers. In an effort to collect data which might suggest areas for curriculum revision and improvement, the study described in this article was conducted by Dr. Russell J. Call, Chairman of the Mathematics Department at Winnacunnet High School, Hampton, New Hampshire, and Mr. Neal A. Wiggin, House Coordinator at Hampton.
Pseudo-Ternary Arithmetic
Authors: Asprey Winifred
Published: November 2020

The kaleidoscopic patterns of the “new mathematics” curricula coupled with the mobility of population in the United States make present-day secondary school classes as heterogeneous in composition as sections of college freshmen. Irrespective of grade or topic, many students react with a shrug and a comment: “Aw, that's old stuff.”
Historically Speaking—: The Bernoulli Family
Authors: Eves Howard
Published: November 2020

There is a general rule to the effect that any given family possesses at most one outstanding mathematician and that, in fact, most families possess none. Thus a search through the ancestors, descendents, and relatives of Isaac Newton fails to turn up any other great mathematician. There are exceptions to this general rule. For example we have, here in the United States, the two Lehmers (father and son) and the two Birkhoffs (father and son). One also recalls the two Cassinis (father and son) of the late seventeenth and early eighteenth centuries, and perhaps one can build a case for the two Clairaut children of the eighteenth century. And of course there were Theon and Hypatia (father and daughter), who lived during the closing years of ancient Greek mathematics. But such cases are relatively rare. All the more striking, then, is the Bernoulli family of Switzerland, which in three successive generations produced no less than eight noted mathematicians.
Classics in Mathematics Education: Sets, Sinners, and Salvation
Authors: Meder Albert E.
Published: November 2020

Author's Note.—”Sets, Sinners, and Salvation” was prepared to describe the context in which the Report of the Commission on Mathematics of the College Entrance Examination Board should be read and considered. It is naturally a great honor, of which I am keenly appreciative, to have it selected in so short a time—seven years—as a classic in mathematical education, but the real classic is the Commission Report. Happily, by an independent decision, the College Entrance Examination Board is reprinting the Report, so that I can again urge, as I did seven years ago, that it be read, reacted to, and tried.
Mathematics in General Education
Authors: Alberty Elsie J.
Published: November 2020

In recent years increasing attention has been given to the significance of mathematics in our culture and to the rapidity with which mathematics has been changing and producing change. As a consequence, there has been much concern both for the content of the mathematics curriculum and for the instructional procedures which characterize the mathematics programs of the schools of our nation. A variety of proposals for developing more effective mathematics programs have been made and considerable research and experimentation has been carried on.
Classics in Mathematics Education: Human Interest Put into Mathematics
Authors: Leacock Stephen , Crosswhite F. Joe
Published: November 2020

The student of arithmetic who has successfully striven with money sums and fractions finds himself confronted by an unbroken expanse of questions known as problems. These are short stories of adventure and industry with the end omitted and, though betraying a strong family resemblance, are not without a certain element of romance.
An Open Letter to Authors and Publishers
Authors: Lasswell Edmund C.
Published: November 2020

Students and teachers of mathematics need more help. Complement your fine texts with short, programmed units.
International Mathematical Education: Mathematical Research and Instruction in Probability Theory
Authors: Engel A.
Published: November 2020

In The education of children the implanting of important mathematical ideas should take place as early as possible. This principle holds especially for the study of chance phenomena. Probability theory is one of the most multifaceted branches of mathematics, and its study requires an introduction to a large number of new and unusual concepts with which pupils need time to become familiar.
Originals in Geometry
Authors: Marsh Harry B.
Published: November 2020

In dealing with originals in geometry, or in any other branch of knowledge or activity, we can safely assume that it is human nature to like originals, to desire to make use of whatever information may be at hand in order to branch out along new and independent lines. This phase of human nature has played no little part in starting the great explorers of all times upon their epoch making expeditions; and who can deny that the discovery of this continent was due to the supreme effort of Christopher Columbus to demonstrate a world-embracing original in spherical geometry. This eagerness for individual achievement and success has been the inspiration of leaders and inventors, and it has been a very effective force back of all progress. It is present to a greater or less extent in all human beings.
The Comprehensive Examination in Mathematics
Authors: Koch Ernest H.
Published: November 2020

My topic, like every other educational problem, is an old one in a new dress. We would make no progress in educational matters if it were not for the fact that our educational systems are undergoing readjustments in accordance with the demands of life.
Are Particular Abilities Necessary for Pupils to Gain an Understanding of the Elementary and Secondary Mathematics as Usually Given at the Present Time
Authors: Wheelock Charles F.
Published: November 2020

The student of mathematics in the elementary school and the high school is called upon to deal with three quite different and distinctly marked kinds of subject matter.
Organizing a Mathematics Laboratory
Authors: Sweet Raymond
Published: November 2020

Why have a mathematics laboratory? There are a number of reasons for operating a mathematics laboratory. One is that the laboratory method of teaching is a pedagogically sound technique. It utilizes an experimental approach that requires each student's participation, but allows him to work at his own rate. This leaves the instructor free to administer more individual help. Also, because the laboratory is usually conducted on a less formal level than the regular classroom, it provides an opportunity for more discussion among the students, and they learn from one another. Most of all, it adds variety as a change from the usual classroom routine.
Scientific Notation
Authors: Callanan Cecelia
Published: November 2020

There is an opportunity in teaching scientific notation to develop anew some appreciation of the beauty of consistency and order in our decimal number system. Scientific notation is introduced in many current texts for junior high school; the writers of the SMSG materials devote a whole chapter to the study of this topic and to its use in connection with the metric system of measures.1
Trees and Unicyclic Graphs
Authors: Anderson Sabra S. , Harary Frank
Published: November 2020

There are many results in mathematics that are both new and elementary. Graph theory, which abounds in theorems of this kind, is the source of an interesting self-contained example which we present in this article.
Five Sailors and a Monkey
Authors: Brashear Philip W.
Published: November 2020

One of the basic problems in the teaching of mathematics originates in the nature of mathematical thinking, particularly in problem-solving situations.
Points and Viewpoints: We Gladly Announce …
Authors: Brune Irvin H.
Published: November 2020

A column of unofficial comment
The Graphing of Absolute Value Inequalities
Authors: Green Thomas M.
Published: November 2020

When dealing with absolute value inequalities one is confronted with the problem of solving a pair of inequalities. For some pairs the solution results in the union of two sets, while for other pairs the solution results in the intersection of two sets.
The Mathematically Gifted Student and Discovery Learning
Authors: Meconi L. J.
Published: November 2020

Numerous articles, books, and research papers deal with both the topic of the gifted student and the topic of a discovery approach to learning. Within the past fifteen years, there has been increasing interest in the mathematically gifted child; and, more recently, mathematics educators have begun to investigate the ways these children learn mathematics.
International Mathematical Education: Some Remarks on Japanese Mathematics Education
Authors: Fehr Howard F.
Published: November 2020

Ever since the results of the International Achievement Tests in Mathematics were announced, much interest has been expressed in the outstanding performance made by the Japanese students.1 The test results displayed an accomplishment that at age thirteen, and overall, was far su perior to the other countries involved. Many analyses have been made of the subject matter involved in the questions, of the type of questions, of the teachers of the students, and of the curricula of each of the countries, to either sustain or disprove popularly expressed superiority of Japanese mathematics education. This article merely reviews the existing structure of Japanese education and points to some unique features of Japanese culture that may offer a partial explanation. It is not intended in any sense to suggest that the Japanese program or its culture— which is evidently satisfying to their own people—is one that would be satisfactory for others. It is merely one of many cultures which we should corne to under stand as the world grows closer knit in its forward progress.
Experimental Programs: Patterns of Characteristics of Successful Mathematics Teachers and Those Who Leave the Profession: A Case Study
Authors: Chapin June R.
Published: November 2020

A Shortage of secondary mathe-matics teachers has existed for many years.1 The supply and demand of mathe matics teachers is difficult to predict on a long-term basis because of the influence of various staff utilization plans, the use of auxiliary personnel, teacher withdrawals, misassignment of teachers, etc., but the shortage trend probably will continue. The public, the government,2 and institutions of higher learning are concerned about the availability of qualified teachers of mathematics for secondary school teaching.
On The Solution of A Problem
Authors: Wilson James W. , Becker Jerry P.
Published: November 2020

Some of the difficulties which students have in developing mathematical problem solving ability are (1) realizing that prob lem solving is more than answer getting, (2) failing to consider exceptional cases, and (3) failing to review the solution in order to (a) validate it, (b) improve it, (c) generalize it, (d) consolidate what has been Iearned, and (e) suggest what else might be learned. Additional difficulties may come front the misleading presentation of a problem.
Vux Triangles
Authors: Cheney Fitch
Published: November 2020

The purpose of this article is to exhibit some of the special properties possessed by a triangle in which the measure of one of its three angles is one-half the measure of another of these angles. The author has named such a triangle a vux triangle, and such a relationship between two of its angles the vux property. (The adjective, vux, derives from the fact that in Roman numerals, V is half of X.)
Gjy insert, The Geometric Continuum
Authors: Fawcett Harold P.
Published: November 2020

The first geometric transformation that had any real impact on my intellectual life occurred in the early fall of 1908 when I was a student in a small Canadian school.
Historically Speaking—: Thabit Qurra and the Pythagorean Theorem
Authors: Shloming Robert
Published: November 2020

During the Dark Ages of Western Europe, Islam was very much mathematically alive. The Arabs were able to solve the most difficult problems of Archimedes and Apollonius at a time when Latin mathematical knowledge was at a level below that of the ancient Egyptians. Indeed, the Arabs were translators and transmitters of mathematical knowledge rather than innovators. Yet the student of mathematics owes them a large debt for preserving a priceless mathematical legacy. For without the contributions of the Arab translators, much of Greek and Hindu mathematics would now be lost. By uniting Greek and Hindu mathematical ideas, the Arabs further developed arithmetic, algebra, geometry, and trigonometry.
Mathematics “In Statu Nascendi,”
Authors: Cummins Kenneth
Published: November 2020

“Mathematics ‘in statu nascendi’—in the process of being invented— has never before been presented in quite this manner to the student, or to the teacher himself or to the general public.” 1 In this very significant statement George Polya invites us to encourage in our classrooms an atmosphere and spirit of creating mathematics. A promising way to do this is to invite, accept, and investigate student suggestions for solutions of mathematical problems or to place the student in a natural or contrived (yet interesting) situation from which mathematics emerges as the student explores possible solutions.
Historically Speaking—: AI-Biruni's Table of Chords
Authors: Bradley A. Day
Published: November 2020

It Is well known that the function employed in Greek trigonometry was the chord of the central angle in a circle of given radius, and that Ptolemy gave a sequence of theorems for computing a table of chords at intervals of 30' to 180° [3].*
Translation of Axes Discovered through the Overhead Projector
Authors: Anderson Frank A.
Published: November 2020

Axis translation has always been a difficult concept for students to grasp. Perhaps the method of approach could be improved. This paper describes a method of approach that has been used by the author and that seems easier than traditional presentations.
The Fascination of Whole Numbers
Authors: Barnett I. A.
Published: November 2020

Before beginning the main topic of my talk, I shall make some remarks and comments that should be of interest to mathematicians and to teachers.
Viewing Diagrams in Four Dimensions
Authors: Hess Adrien L.
Published: November 2020

When a figure in three-space is represented by a drawing on a sheet of paper, which is essentially two-space, the figure is usually distorted. As a result, some people have trouble visualizing three dimensions from such a representation.
To Tip a Waiter— a Problem in Unordered Selections with Repetitions
Authors: Dessart Donald J.
Published: November 2020

In Most elementary textbooks of algebra or probability intended for use in the secondary schools, the topic of unordered selections of objects, more commonly known as combinations of objects, is usually studied.
Nim and Natural Numbers
Authors: Merriell David
Published: November 2020

There is a well-known strategy for winning the game of Nim.1 The strategy involves the use of an unfamiliar operation in the set of natural numbers. In this article we describe the strategy. Then we examine the unfamiliar operation, giving special attention to its connection with the theory of groups.
The 1969 Advanced Placement Examinations In Mathematics-Complete And “Unexpurgated”
Authors: Finkbeiner Daniel T. , Neff John D. , Williams S. Irene
Published: November 2020

Under the direction of the College Entrance Examination Board the Ad-vanced Placement Program in Mathemat-ics has evolved in content as steadily as it has grown in size. (See table 1.) Course descriptions are under constant review and are revised biennially by the Com-mittee of Examiners to keep pace with trends in school and college mathematics instruction. This gradual evolution can be detected by comparing successive edi-tions of the course descriptions from 1955 to the present. The most significant change in content occurred in 1968-69 when a choice of two courses and two examinations (Calculus AB and Calculus BC) was instituted [1].*
Some Suggestions for Motivating Students in General Mathematics Classes
Authors: Bachman Alfred M.
Published: November 2020

The problem of motivating student in high school general mathematics classes is of great concern to a majority of high school mathematics teachers. In the innercity schools of our large metropolises this problem is compounded in that the school is located, and that the students reside, in ghetto neighborhoods. Students coming from conditions of poverty and despair are seldom motivated toward high performance in academic work. This report describe an attempt to seek way of motivating these students.
Geometric Construction: The Double Straightedge
Authors: Wernick William
Published: November 2020

THE classical Euclidean constructions allow only the unmarked straightedge and a pair of compasses, now commonly called “a compass.” With these limited tools we can carry through the many beautiful constructions of elementary geometry. A lack of understanding of the “rules of the game” leads to such confused and incorrect statements as “It is impossible to trisect an angle.” But of course we can trisect a right angle; and as for trisecting any angle, there is a simple and elegant construction by Plato that does this very nicely, if we allow such a slight concession as two marks on the straightedge, or even one mark and the endpoint!
The Limit Concept on the Geoboard
Authors: Harkin J. B.
Published: November 2020

Archdiedes exploited the method of exhaustion to determine the area of a circle and the value of π. By inscribing and circumscribing regular polygons of 2n sides (see fig. 1) the interior of the circle contain the interior of the in cribed polygon and is contained in the interior of the circumscribed polygon. The corresponding measures of the interiors (areas) satisfy the following: measure of interior of inscribed polygon < measure of interior of circle < measure of interior of circumcribed polygon. However, the computational task in calculating the measure of the interiors of polygons is formidable if the traditional expression is used: A = ½ap, where a = apothem, p = perimeter.
New Products
Authors: Hansen Viggo P. , Johnson Randall E. , Koetke Walter J.
Published: November 2020

The unit or minicours in logic so frequently favored in high schools has the laudable objective of familiarizing the student with the format of rigorous thinking. This, however, is but the bare though necessary skeleton of daily thought. Pushing the metaphor a bit, the nerve ends of the thinking person must analyze the emotional aura surrounding the ideas expressed by others. In a word, the thinking person must be able to distinguish between the form of thought (logic) and the dressing of thought (propaganda). This game, Propaganda, addresses itself to clear thinking, so that the student becomes familiar with the six major techniques of swaying the thinking of people. Each technique has at least, eight and at most ten varieties. The players (there must be two, but. better if there are more) first, learn how they fool themselves; then, they proceed to techniques of influencing others. Given concrete situations. the players seek to identify the precise variety of technique used in the propaganda statement. Twenty plays constitute a game. The winner is classified as a “clear thinker” for that, round; the loser may be anything from a “ding-a-ling” to a “Gullible Gus.” The explanation book is quite clear. The authors, Robert W. Allen of Nova School and Lorne Greene of “Bonanza” (!), have created a most useful tool for thinking one's way through the porridge of propaganda.— Barnabas B. Hughes, San Fernando Valley State College, Northridge, Calif.
Instant Insanity: That Ubiquitous Baffler
Authors: Duncan Dewey C.
Published: November 2020

This discussion is occasioned by two notes in the October 1969 issue of the Mathematics Teacher, one a letter to the editor from Barbara Almli on page 446, and the other Philip Peak's review on page 477 in his “Have You Read …?” section of T. A. Brown's “A Note on ‘Instant Insanity’” (Mathematics Magazine, September 1968, pp. 167-69). Each of these items describes the puzzling challenge known as “Instant Insanity.”
A Fortran Program for Computer Plotting of Functions of Two variables
Authors: Krabill David , Long Clifford
Published: November 2020

In Teaching three-dimensional geometry or in studying functions of several variables, it is quite convenient to have graphs of functions of two variables readily available. While digital computers equipped with plotters are ideal for providing these graphs, the cost of such plotters keeps them from being as accessible as the computer itself.
An Algorithm for Partial-Fraction Expansion
Authors: Rogers Joseph W. , Rogers Margaret Anne
Published: November 2020

We Usually expand a rational function by first reducing it to the sum of a polynomial and a proper fraction that is the quotient of two polynomials in which the degree of the numerator is less than the degree of the denominator. We expand the fraction by writing it as a sum of partial fractions with undetermined numerator coefficients.
Trigonometric Formulas Via Areas
Authors: Greitzer Samuel L.
Published: November 2020

For several decades mathematics educators have been on the lookout for a “unifying concept” that would transform mathematics from a sequence of compartmentalized subject areas into one unified whole. Once an educator has decided that he has found the one unifying concept. he rewrites secondary school mathematics to conform to this concept. The philosophy seems to be that if a mathematical development can be made to conform to the unifying concept selected, it must be made to do so.
The Quadratic Formula—an Enrichment Approach
Authors: Stilwell Kenneth
Published: November 2020

The procedure of completing the square, used in deriving the quadratic formula, is pervasive in mathematics. The following is presented as an alternate method of derivation of the quadratic formula for students studying analytic geometry. It is not intended to replace the traditional derivation, but rather is presented as an enrichment topic based on the student's ability to solve pure quadratic equations (that, is, equations of the form ax2 + b = 0, a ≠ 0) and perform a simple transformation.
The Forum: Individualization of Instruction: Individualizing Mathematics in a Modular-Scheduling Program
Authors: Indelicato Arthur
Published: November 2020

Since the advent of the formal class room, teacher have been struggling with the problem of meeting the individun needs of their students.
The Traveling Salesman Problem
Authors: Kennedy Joe
Published: November 2020

A COMPANY with branches in ten cities hires a man to visit each branch office once to install pencil sharpeners. They may hire him in any of the ten cities and his employment is to be terminated in the tenth city. Where should he be hired and what route should he take to make his total travel a minimum? (The company does not pay his way home.)
New Programs
Authors: Hight Donald W. , Kansky Robert J. , Richards Don K.
Published: November 2020

A mathematics team at DeWitt Junior High School, Ithaca, New York, recognized that although the previously developed seventh-grade curriculum was satisfactory, the individual needs of the students were not fully met by classroom presentation. This team used the existing curriculum to devise an individualized program that would (1) encourage individual progress, (2) avoid excessive imposition of hardware on the learning process, and (3) maintain and encourage teacher-pupil interaction.
Areas by Infinite Series
Authors: Morgan Lawrence A.
Published: November 2020

Archimedes discovered a way of finding the exact area of plane surfaces hounded by curved lines without integral calculus.1 The method consists of fitting simple figures like squares and triangles into the curve. The spaces between the curve and the simple figures are gradually filled up with smaller and smaller figures. If the areas of the simple figures can he calculated along with their infinite sum, the desired result is obtained. Two examples follow.
Some Lessons Learned by SMSG
Authors: Begle E. G.
Published: November 2020

In the more than fourteen years since the founding of the School Mathematics Study Group, we have learned a good deal about mathematics education. A review of some of this may be of interest. Let me start with two obvious facts, both of which we all probably knew before SMSG got started.
A Guide to Mathematical Discovery
Authors: Ocrant Ian
Published: November 2020

Introductory Note. Teachers are often told to teach students by the discovery method. As a junior student at George Washington High School, the writer of this article used the discovery method as described here while participating in a Computer Mathematics Curriculum Project sponsored by the National Science Foundation and directed by William Dorn and Ruth Hoffman of the University of Denver. Ian was asked to document his thinking in a diary fashion so that others could see, firsthand, the discovery method at work. Irwin J. Hoffman, George Washington High School, Denver, Colorado.
Griefless Graphing for the Novice
Authors: Andersen Harold
Published: November 2020

Typically, a student in the first course of high school algebra finds graphing obscured by a hassle of “cleaning up” equations. While the skill of manipulating algebraic quantities is important, it detracts from the important aspects of graphing.
Using Algebra To Solve An Interesting Card Trick
Authors: Crouse Richard J. , Reese Clyde
Published: November 2020

Most mathematics educators would probably agree that an effective strategy for teaching mathematics should strive to meet at least the following criteria. The strategy should (1) catch the attention and interest of the students, (2) maintain this interest and attention, and (3) fulfill the objectives of the lesson. The following strategy fulfills these conditions.
The Classical Cake Problem
Authors: Nelson Norman N. , Fisch Forest N.
Published: November 2020

Maintaining student interest and enthusiasm are ubiquitous problems that confront all teachers. The following paragraphs identify a geometry-related problem that can evoke considerable interest, enthusiasm, worthwhile study, and satisfaction from your students.
A Metric World
Authors: Straley H. W.
Published: November 2020

Students of secondary mathematics are almost always confronted with what is reasonable. It is sometimes worthwhile to shake them up with something that is not so reasonable. One way to do this is to modify a definition and see what effect this new definition has on some conventional ideas.
A Computer Graphics System for Polynomials
Authors: Shneiderman Ben
Published: November 2020

The development of mathematical intuition is a fundamental goal of a well-designed course in mathematics. The superior instructor teaches more than operations or rules of transformation; he seeks to convey a deeper understanding of the subject matter, which enables a student to have insight and to derive new results. Often, the complexity of a problem prevents the instructor from making such a presentation, and the student fails to grasp the complex interaction of the components of the problem.
Recreation: Tic-Tac-Toe in Polar Coordinates
Authors: Browne Joseph B.
Published: November 2020

The game of tic-tac-toe has long been used as a motivational device in mathematics instruction. Students can play by using numbers to create magic squares (Jones 1951), they can study the strategy (Brumfiel 1963; Ruderman 1951), they can create new strategies by playing to lose (Cohen 1972), or they can try to generalize the game to larger arrays (Ruder-man 1951) or three dimensions (Mercer and Kolb 1971). Kenneth Hummel invented a variation to help introduce point naming in polar coordinates. The game is described below using radian measures for angles. With younger students, degree measure is suitable too.
On Simpson’s Rule
Authors: Monzingo M. G.
Published: November 2020

Simpson’s rule is encountered in practically every sequence of calculus courses. Most students seem to feel that an application of Simpson’s rule should yield a better approximation of a definite integral than the use of either inscribed or circumscribed rectangles. They do, however, seem to wonder about the convergence of Simpson’s approximation to the actual value of the definite integral. This note shows that such is the case, using elementary manipulations within the grasp of beginning calculus students.
Nonverbal Teaching
Authors: Mon Gordon R.
Published: November 2020

At a time when traditional teaching techniques are being seriously questioned and alternatives—curricula, content, materials, and even structures and styles of education—are being tried, tested, and heatedly debated, next to nothing is said about saying nothing. No one questions a teacher’s foremost means of conveying information—talking.
What Does “Everyman” Really Need from School Mathematics?
Authors: Bell Max S.
Published: November 2020

Charting new directions for mathematical literacy and competency, for our future citizens. A “must” reading for every mathematics teacher and curriculum developer.
The Logarithmic Spiral
Authors: Maor Eli
Published: November 2020

This article will be of interest to teachers of analytic geometry. It points out some interesting properties of the logarithmic spiral and cites two examples from physics where this spiral finds applications.
On Fermat's Quadrature of the Parabola
Authors: Zatzkis Henry
Published: November 2020

The reader should find this a refreshing exercise in discovery.
Tetrahedral Frameworks
Authors: Trigg Charles W.
Published: November 2020

A model for analysis of polyhedral frameworks.
Paper Folding and Convergent Sequences
Authors: Rapaport William J.
Published: November 2020

Paper folding can help in understanding some infinite sequences and in finding their limits. A simple physical model useful at all levels of ability is presented and infinite sequences of interest to senior high school students are explored.
Solving the Equation X2 + 1 = 0
Authors: Spezeski William J.
Published: November 2020

The author discusses some interesting solutions of the equation x2 + 1 = 0. Application occurs in the modulo systems, and in contrast the author includes solutions obtained using n × n matrices.
Variations on a Theme by Polya
Authors: Dacey Raymond
Published: November 2020

An example of a mathematical discovery that leads to a significant generalization. It can be used to show students how to derive a surprising relation.
The Objectives of Mathematics Education
Authors: Greenberg Herbert J.
Published: November 2020

Here are some questions and criticisms that need to be answered before the mathematics curriculum is changed. The questions are also relevant for day-to-day decisions in the classroom.
Collaboration in the Mathematical Community
Authors: Bagnato Robert A.
Published: November 2020

Belief is emphasized as a necessary ingredient along with intuition and rigor in the introduction to the universe of mathematics.
Geometry is more than proof
Authors: Hoffer Alan
Published: November 2020

Each year we ask many of our first-year students at the University of Oregon to list the mathematical subjects or topics that they liked best and topics they liked least in their precollege classes. Although several subjects were “favorites,” the subject that was almost universalJy disliked was geome- try in high school.
Applications: International Standard Book Numbers
Authors: Knill George
Published: November 2020

In the years 1968-72 all books through- out the world began to receive coded ten-digit numerical labels called International Standard Book Numbers (ISBNs). The ISBN functions as an unambiguous means of ordering or maintaining inventories of books, especially when records are stored by computer. The ISBN is assigned by publishers to their books.
Some Problems with Fractions for the Middle School
Authors: Lappan Glenda , Winter Mary Jean
Published: November 2020

The National Assessment of Educational Progress (1980) scores confirm that fractions, percentages, and decimals are among the topics that middle grade students find difficult. Exploring interesting mathematical questions whose solutions require computations involving these concepts is an excellent motivation for improving basic understandings and skills. In this article we will present a series of related questions that involve fractions, percentages, and decimals. The solutions to these questions can be found by paper-and-pencil work, or after the students have gained sufficient understanding, the questions can be answered by using a computer or programmable calculator. Answers appear in Appendix A.
Applications: Mathematics in Forensic Science
Authors: Knill George
Published: November 2020

Knowing the exact physical dimensions of a victim of a crime is extremely useful in identifying the victim. When a skeleton is found, a forensic scientist uses the lengths of certain bones to calculate the height of the living person. The bones that are used are the femur (F), the tibia (T), the humerus (H), and the radius (R). See figure 1. When the length of one of these bones is known, one of the following formulas is used to determine the height. All measurements are in centimeters.
Gambling, Casinos, and Game Simulation
Authors: Heiny Robert L.
Published: November 2020

ln 1973, the University of Northern Colorado (UNC) initiated a new undergraduate general education program for the purpose of encouraging innovative courses and interdisciplinary cooperation. The university has never required mathematics as a part of its general education program. However, the Mathematics Department at UNC is becoming more involved with service courses for other departments. In the hope of encouraging more students to take some mathematics, I developed a course entitled “Gambling, Casinos, and Game Simulation.” This article explains the objectives, the content, and the intent of the course.
Clubs: Chi Alpha Mu Junior Mathematics Clubs
Authors: Shipley Lois Q.
Published: November 2020

Chi Alpha Mu is the national mathematics club for junior high and upper-middle-school students. The Greek letters stand for Creative Adventures in Mathematics-the goal of the organization. Chi Alpha Mu is the brainchild of Sarah M. Burkhart, who initiated the program in three schools in Tulsa, Oklahoma, in 1967. By 1978 there were sixty-five clubs in about a dozen states and nearly sixty-three hundred members. The group affiliated with NCTM about that time. Sarah Burkhart's death last April left a void, but Jack Skelton, Instructional Assistant for Secondary Mathematics, Tulsa Public Schools, P.O. Box 45208, Tulsa, OK 74145, has agreed to coordinate the administrative affairs of the national office. Inquiries to Jack Skelton will be answered with a public-relations kit providing information on goals, rules, dues, and benefits.
Some Irrational Results with Irrational Numbers
Authors: Dolan Daniel T.
Published: November 2020

Approximation pitfalls when working with calculators and computers.
Some Misconceptions Concerning the Concept of Variable
Authors: Rosnick Peter
Published: November 2020

Are you careful about defining your variables?