bluuebunny/crossref_metadata_2025 huggingface dataset:
On the rapid sketching of plane parametric curves
Authors: Greenspan Donald
Published: November 2020

Curve sketching is a valuable visual aid for studying the properties of functions via algebraic techniques.
Mathematics instruction via television
Authors: Alkire G. Don
Published: November 2020

To meet the needs of increased enrollments and to provide for in-service training, institutions are turning to courses offered via television. This report illustrates one technique used to evaluate such courses.
Benjamin Franklin and mathematics
Authors: Feldmann Richard W.
Published: November 2020

Although Benjamin Franklin (1706-1790) was one of America's foremost inventors and natural philosophers, he cannot be classified as a great mathematician. His belief that the only value of science is its practical side is shown by his statement, “Nor is it of much importance to us to know the manner in which nature executes her laws.
The theory of braids
Authors: Artin Emil
Published: November 2020

Professor Artin's famous lecture is reprinted because the Editor feels secondary teachers will be interested in a paper which is easily understood and uses some very important mathematical concepts.
Historically Speaking,— Benefits Derived from Mathematics Projects
Authors: Fish Dorothy
Published: November 2020

Students in our high school, enrolled in any elective mathematics courses, are encouraged to undertake mathematics projects, which are then entered each spring in the annual District Science Day Contest sponsored by the Ohio Academy of cience. Winners in the District Contest are eligible to compete for awards in t he tate Science Day Contest. This contest creates great interest and enthusiasm in our school, and to win a “superior” rating in the State Contest is considered one of the highest honors that a student can achieve.
“To hold, as’ twere, the mirror up to nature; to show the very age and body of the time.”
Authors: Langer Rudolph E.
Published: November 2020

Among the challenges of the day, none is being sounded more insistently than the call for more, and more effective, teaching of mathematics.
Historically Speaking,— Gergonne, founder of the Annales de Mathématiques
Authors: Guggenbuhl Laura
Published: November 2020

Joseph Diez Gergonne was born on June 19, 1771, in the parish of St. Roch at Nancy, just a few years after this city had been transformed into a center of artistic elegance. Indeed, his father had been one of the artists who had had a part in the creation of this wonderland of lavish grace.
Historically Speaking,—: Notes on the centroid
Authors: Court Nathan Altshiller
Published: November 2020

The term “centroid” is of recent coinage. It is used as a substitute for the older term “center of gravity,” when the purely geometrical aspects of that point are to be emphasized. The term is peculiar to the English language. The French use “centre de gravité” on all occasions, and others use terms of similar meaning.
Mathematics and industry—a report
Authors: Wilcox Marie S.
Published: November 2020

Local and state councils of teachers of mathematics affiliated with the National Council report many and varied instances of co-operation with industry.
An introduction to the angle measurement theorems in plane geometry
Authors: Schor Harry
Published: November 2020

The usual approach to the angle measurement theorems is static. The students are given a central angle in a circle and they compare its degree measure with the degree measure of the intercepted are.
Tips for Beginners: What do they really say?
Authors: Mehl William G.
Published: November 2020

Modern textbooks are doing more to help in developing understanding of concepts in mathematics. But classroom teachers can enhance the work of the purposeful and sincere textbook author by their presentations in class. Any textbook which stresses methods of solving simple equations can be readily appreciated.
Roots of the cubic
Authors: Yates Robert C.
Published: November 2020

“A discriminant and a mechanical arrangement to determine the real zeros of a cubic”
A triangle construction
Authors: Scholomiti N. C. , Hill R. G.
Published: November 2020

Constructing a triangle when given the lengths of the three perpendicular bisectors of the sides
The velocity of escape
Authors: Stretton William C.
Published: November 2020

Physics for the modern world in the mathematics class
Experimental Programs: A two-year study of eighth-grade Algebra I
Authors: Hegstrom William J. , Riffle Donald E.
Published: November 2020

Ability grouping for eighth-grade Algebra I is not a new idea.1 It has been tried many times.2 During the school years 1960-61 and 1961-62 a program of this nature was carried out at Delray Junior High School, Delray Beach, Florida.
Mathematics and computing in high school: a betrothal
Authors: Forsythe Alexandra
Published: November 2020

One does not need either a computer or a crystal ball to see that many changes must be made in the high school mathematics curriculum to take account of electronic digital computers.
A unit in high school geometry without the textbook
Authors: Avers Paul W.
Published: November 2020

A step toward discovery teaching
Current reforms in the mathematics curricula—a passing phase or progress?
Authors: Ferguson W. Eugene
Published: November 2020

The goal is excellence.
Linear Indeterminate Problems
Authors: Fogo James D.
Published: November 2020

A systematic approach to the solution of many equations resulting from attempts to solve some puzzle problems
Historically Speaking,—: Myth, the Muse, and Mathesis
Authors: Boyer Carl B.
Published: November 2020

The October, 1962, issue of The Mathematics Teacher carried four main articles, all concerned with changes which have taken place and are taking place in mathematics. Dean Mina Rees, in the opening article, “The Nature of Mathematics,” pointed out that our subject had changed radically in the last century, and that even today there are earnest controversies concerning its future.
The Role and Nature of Mathematics
Authors: Hammer Preston C.
Published: November 2020

Some frank comments addressed to teachers of mathematics
Approximating an Angle Division by a Sequence of Bisections
Authors: Pursell Lyle E.
Published: November 2020

A fresh look at a classical problem
Integers which are differences of squares
Authors: Slook Thomas H.
Published: November 2020

An interesting existence proof
An introduction to solving problems
Authors: Topoly William
Published: November 2020

In solving a first-degree equation with the variable on one side of the equation, a student must rely on deductive reasoning to find the solution. For a given equation, there may be many different methods of solving.
How for is it from here to there?
Authors: Fischer Irene
Published: November 2020

Man's ancient activity still presents problems. Who will solve them?
Reflexive, symmetric, and transitive properties of relations
Authors: Hoy Dorothy H.
Published: November 2020

Some contemporary ideas graphically illustrated
A scale for “scaleneness”
Authors: Rosenthal Evelyn B.
Published: November 2020

A novel rating for not-so-novel triangles
Some Guidelines for Selecting a Programed Text in Mathematics
Authors: Kalin Robert
Published: November 2020

There are now over one hundred programed texts available for use in mathematics classes from early elementary school through college. The purpose of this article is to supply mathematics teachers with guidelines for deciding whether a text is suitable for his class.
Two Theorems Concerning Partial Fractions
Authors: Just Erwin , Schaumberger Norman
Published: November 2020

The representation of rational expressions in terms of partial fractions has important uses in many branches of mathematics. Two relatively unknown, but nevertheless useful, properties of partial fractions will be considered in this article.
An Approach to Euclidean Geometry through Vectors
Authors: Szabo Steven
Published: November 2020

Among the central ideas in mathematics today is the notion of a vector space. This concept has many applications in algebra, geometry, and analysis. In addition to its applications, the concept of a vector space can serve to relate the study of algebra and geometry in the secondary school mathematics curriculum. For the most part, the studies of algebra and geometry in the high school curriculum are not at all related. In fact, it is the case in many instances that the study of geometry turns out to be merely a strange interlude between the study of algebra in the ninth grade and the continued study of algebra in the eleventh grade.
Experimental Programs: An Investigation of the Effect of Types of Exercises on Mathematics Learning
Authors: Bassler Otto C.
Published: November 2020

An experimental study was carried out by the University of Maryland Mathematics Project (UMMaP)* to investigate the effects of some instructional variables in mathematics learning. Specifically, the two instructional variables being tested were (a) immediate knowledge of correctness of response; and (b) two distinct types of exercises: those framed in a physical world setting and those framed in a purely mathematical setting.
Strategies for Teaching Concepts by Using Definitions
Authors: Ginther John L. , Henderson Kenneth B.
Published: November 2020

Reflection indicates that there is a kinship between terms which designate, and concepts. A teacher would do much the same thing if he were teaching the concept of a rectangle as if he were teaching the meaning of the term rectangle. Accepting the risk of oversimplification, we suggest that a term is a name of the concept denoted by the term. It follows that defining a term is one way of teaching the concept denoted by the term.
Even More on Pascal's Triangle and Powers of 11
Authors: Low Leone
Published: November 2020

At the risk of belaboring the point, I feel that a further comment on (1) and (2) might suggest a way to use Pascal's triangle to help the student to understand the binomial theorem and place notation. The more skillful pedagogues might use the discovery technique in the presentation of the material.
Grammarmathically Speaking
Authors: Jamison King W.
Published: November 2020

Much has been said which identifies mathematics as an art. Few students, teachers of mathematics, and laymen would deny the esthetic value of mathematics, and tradition labels mathematics as an art. As one example, the curriculum of the medieval university included mathematics among the seven liberal arts. Mathematics has also been thought of as a science for a long time. Now, aided by modern communications, even the most uninterested person associates mathematics with science and scientists. It is therefore comparatively easy to accept the notion that mathematics is an art and that mathematics is a science.
International Mathematical Education: High School Mathematics Reform in Greece
Authors: Glavas C. B.
Published: November 2020

In this article the school mathematics reform in Greece is examined. The exposition is on the present phase of the reform, which is limited to the lower cycle of the high school (ages twelve to fifteen).
Historically Speaking—: The Position of Thomas Carlyle in the History of Mathematics
Authors: Wursthorn Peter A.
Published: November 2020

I saw David Brewster the other day, who received me kindly and spread out his bank draft for fifteen guineas like a man. He told me further that a translation was for certain to be set about and that I as certainly should have the first offer of it. The work is a thing I can work at if the “gea of life” be in me at all, and for that cause alone I propose to accept.
Some Ideas on the Study of Geometry
Authors: Smith Eugene R. , Gaylord Harry D. , Babb Maurice J. , Breckenridge William E. , Shultz Charles R.
Published: November 2020

In the preparation of this paper I have followed certain suggestions on the study of Geometry contained in the Report of the Committee of Fifteen on a Geometry Syllabus. The introduction to this report gives an historical account of various attempts to reform the teaching of geometry. This section should prove, therefore, not only interesting but extremely valuable to students and teachers of mathematics. From it we gleam some very noteworthy facts in regard to the progress that has been made in the attempts to place the study of geometry on a sane and satisfactory footing in the school curriculum.
The Mathematics League
Authors: Paarlberg Teunis J.
Published: November 2020

A new sport is gaining in popularity among high schools in the New England states. It has all the suspense and color of a real athletic contest, but brain—not brawn—is the deciding factor.
Life and Times of Johann Kepler
Authors: Tuck Bernard H.
Published: November 2020

The year 1571, when Johann Kepler was born, had little to mark it as exceptional in a time when events moved but little faster than in ages past. The shadow of the Dark and Middle Age still spread the heavy, restraining Cloak of authority over the live of people.
Gambling Doesn't Pay!
Authors: Sterrett Andrew
Published: November 2020

A gambler with a large bankroll who bets a dollar every minute at a roulette table in America can expect to lose $1,000 in 13 days, 4 hours, and 40 minutes. At a slightly different roulette table in Europe, he can expect to take almost double the time to lose his $1,000.
Experimental Programs: An Experimental Study of Patterns of Presenting Transitivity Statements
Authors: Holtan Boyd
Published: November 2020

The concept of mathematical inequalities is now found in most modern secondary mathematics text materials, and often in elementary school programs. The unit on mathematical inequalities usually considers the transitive property of the equality and inequality relations.
Experimental Programs: An Experimental Twelfth-Grade Mathematics Course
Authors: Gabai Hyman
Published: November 2020

A Current problem in high school mathematics education concerns the question of what mathematical topics ought to be taught to twelfth-grade high school students.
The N-Sectors of the Angles of a Square
Authors: Smart James R.
Published: November 2020

The three bisectors of the angles of a triangle are concurrent at a point called the incenter. The mathematics developed in this article results from experimenting with ideas suggested by the theorem about the bisectors. The conclusions depend only on elementary geometry, but they are interesting in their own right.
Two Combinatorial Theorems
Authors: Glicksman A. M. , Ruderman H. D.
Published: November 2020

Although the primary purpose of this article is to exhibit and prove two interesting relationships involving the binomial coefficients that we have not encountered elsewhere,1 we have several subsidiary objectives. We endeavor to show how the relationships arose naturally in the course of solving other problems; we attempt to strip them of the usual aura of mystery that often surrounds a nontrivial mathematical result; and we encounter an aspect of mathematical induction not often mentioned to students, nor even known to many teachers of mathematics. In this way we hope to endow the article with heuristic as well as mathematical content.
Historically Speaking—: Brachistochrone, Tautochrone, Cycloid— Apple of Discord
Authors: Phillips J. P.
Published: November 2020

The curve generated by the motion of a point on the circumference of a wheel rolling on a flat surface is the cycloid, also known for its mechanical properties as the brachistochrone and the tautochrone, and also described by one mathematics historian as an apple of discord and by others as the Helen of geometry—beautiful, but the source of warfare.
The Majesty of Numbers
Authors: Bradfield Donald L.
Published: November 2020

From the magnificent worlds of the finite and infinite emerges the concept of numbers. Since the origin of all materials things and the beginnings of humanity, man has sought to improve and refine his concepts of numbers. Historically, numbers have been thought to have three properties—magnitude, order, and shape. In modern mathematics, cardinality and ordinality are the terms applied respectively to these first two properties.
Classics in Mathematics Education: Five Decades of Mathematical Reform— Evaluation and Challenge
Authors: Betz William
Published: November 2020

The present year has occasioned numerous reviews of the march of events since the turn of the century. Is it not fitting that a similar scrutiny should occur in the field of mathematical instruction? And so, it is a pleasure to be able to call attention to the Anniversary Book, now rapidly approaching completion, of the Central Association of Science and Mathematics Teachers. It is entitled A Half Century of Teaching Science and Mathematics. In like manner, it is now thirty years since the National Council of Teachers of Mathematics was organized.
Historically Speaking—: The Peruvian Quipu
Authors: Diana Lind Mae
Published: November 2020

Man, having curiosity as one of his basic characteristics, has long desired to unlock the secrets of those who have gone before him in time. Thus, he has embarked on archeological expeditions, has studied history books, has consulted ancient sources of information, and has generally set about looking for the answers to his questions.
Solving Problems in Geometry by Using Complex Numbers
Authors: Garfunkel J.
Published: November 2020

Someone once said that mathematics consists of finding new solutions to old problems and old solutions to new problems. Sometimes, however, it becomes necessary to revive old solutions for old problems. Approximately a hundred years ago C. A. Laisant and other mathematicians of that time used vector methods, with coordinates in the complex plane, to solve certain types of problems in geometry involving polygons having the same centroid, called concentric polygons. This method has been neglected, and it is almost forgotten today. It is the purpose of this article to attempt to revive this method by illustrating its effectiveness in solving certain types of problems in elementary geometry.
Classics in Mathematics Education: Mathematics as a Fine Art
Authors: Shaw James B.
Published: November 2020

Mathematics studies a vast world of objects and their transformations, their static and dynamic characters, intricate and tangled, yet systematic and ordered to a degree far greater than that of any other known world. This world has been created by the mind of man, partly in order to enable him to handle the world of phenomena, but mostly for the elegant beauty of its objects.
In-Service Education in Mathematics: In-Service Education in British Columbia
Authors: Clark J. F.
Published: November 2020

Take a province with an area 90,000 square miles larger than the state of Texas. Sprinkle 1.8 million people across it. Put 0.4 million pupils in its public schools. Add 16,000 teachers. Then replace every mathematics course from Grades 1 to 12 in the five years from 1962 through 1966. British Columbia is where the action is. Or—is it just another area where the action should be?
Reviews of Films
Authors: Coxford Arthur F.
Published: November 2020

Of Stars and Men employs animated charac ters and a beautiful music track (Bach) in ex plaining man's position in the universe, That man, not the lion, is king of the animals is as serted in the initial phase of the film. But as man begins to probe the universe in which he lives, he finds that he is not the center.
The Recursion Formula
Authors: Garfunkel J.
Published: November 2020

In Trying to find the sum of the fractions and , we usually replace each fraction by an equivalent fraction, e.g., . What is the purpose of this simple illustration? It shows clearly how mathematicians work. Given a prob lem, they try to solve it by solving an easier problem.
“b-ary” Fractions
Authors: Wiscamb Margaret
Published: November 2020

When we teach numeration Systems in bases other than ten, we usually con-fine our discussion 1o the positive integers. The student with an inquiring mind (and I hope you are fortunate enough to have one in every class) may see avenues for further exploration and ask questions such as these: How about decimal fractions—can we change them to some other base? Can we have “decimal” fractions in bases other than ten?
Use of Directional Derivative in Locating Extrema
Authors: Stretton William C.
Published: November 2020

It Is possible to use the directional derivative effectively in locating relative extrema of a function of two variables. Why this method has been virtually over looked is a mystery. The criteria given in most texts, if given at all, for testing whether a critical point is a maximum or a minimum are usually developed from a Taylor's series for functions of two vari-ables. It has been my experience that most first-year calculus students get lost some where in this development. The method I am going to propose employs methods used in functions of a single variable and hence is more meaningful, I've found, with such students.
Reviews of Films
Authors: Coxford Arthur F.
Published: November 2020

The film begins with a talk by a representative of the Ford Motor Company. The emphasis in this introduction is on the role research has played in making possible many remarkable ad vances in science. The introductory remarks are followed up by scenes depicting research results being used by technicians.
A Dialogue on Inverse Functions
Authors: Leetch J. F.
Published: November 2020

The use of dialogue to present mathematical ideas has been illustrated recently in a collection by Alfréd Rényi, 1 and in several of the Mathematicall Association of America's Calculus Film Series, such as The Theorem of the Mean (Policeman).2 This author has found such presentations to be appealing and to have a lasting effect. The following dialogue has been motivated by these presentation and by the desire to present plausible difficulties in the study of inverse functions. Certain aspects of the dialogue, however, are not considered very plausible.
Computer-oriented Mathematics: An Eulerian Development for Pi: A Research Project for High School Students
Authors: Smithson Thomas W.
Published: November 2020

During a summer's study under the auspices of the National Science Foundation,1 our History of Mathematics class encountered a most interesting Eulerian problem in infinite series. The problem is exemplary of an abundance of wonderfully rich and exciting historical materials. Acquiring a background in the history of mathematics is important to the high school teacher, for it provides an invaluable source of problem materials with which to capture the imagination and intellectual energies of students. High school mathematics curricula have often been criticized for not providing students with the opportunity to experience the excitement and the deep creative satisfaction of original se]f directed mathematical research. It is hoped that the development that fo1lows provides an example of a viable mathematics research project for secondary school students.
Motivating Deduction
Authors: Sitomer Harry
Published: November 2020

It Is commonly believed that the ability to recognize validity or invalidity in a proof should be part of the intellectual equipment of all, certainly of college students. But there is general disagreement on what constitutes valid proof. Induction is widely accepted as a method of proof if the number and variety of supporting instances are considered reasonable. Scientists will accept induction, but they want a larger number of instances, and they are serously disturbed by even one counterexample. Mathematicians, however, are not satisfied by any but a deductive proof. As teacher of mathematics we try to make a convincing presentation for deduction. Unfortunately we don't succeed as often as we would like. This was brought home forcefully to the writer when a student in a college generalmathematics class, asked to prove that the sum of two odd integers is an even integer, offered the following (the format of which is preserved).
The Polygroup Technique
Authors: Allen Ernest E.
Published: November 2020

High school mathematics teachers are frequently criticized for a lack of variety in their teaching methods. The typical class period consists of an examination of homework, then an introduction of new material (usually by lecture), followed by supervised study. An alternative to this routine is suggested in the following paragraphs.
A Non-Field
Authors: Wiscamb Margaret R.
Published: November 2020

Often much can be learned from the study of a mathematical system that, while not a field, nevertheless has many interesting properties.
A Surveyor's Problem
Authors: Pederson Arnold
Published: November 2020

A Surveyor recently came to our high school in quest of some help in solving a most interesting problem. The problem has since served as a fine application for my trigonometry students. It involves some knowledge of geometry, trigonometry, and algebra.
New Products
Authors: Hansen Viggo P. , Johnson Randall E. , Koetke Walter J.
Published: November 2020

This series can be used to supplement most general-mathematics textbooks. The teaching of general mathematics at the senior high school level can be difficult for a variety of reasons. Difficulties include the wide range of computational ability of the students, poor attendance, lack of relevancy from the student's point of view, the lack of good instructional material that meets the wide range of ability, and the lack of adequate visual aids that can be used on an individual basis.
Using Synthetic Division by Quadratics to Find Rational Roots
Authors: Hutchinson Margaret Wiscamb
Published: November 2020

This paper presents a simple algorithm, similar to synthetic divison, for dividing a polynomial by a quadratic polynomial. in an interesting application of this algorithm, we can significantly reduce the labor involved in searching for the rational roots of a polynomial equation.
Circular Coordinates: A Strange New System Of Coordinates
Authors: Trask Frederick K.
Published: November 2020

All too often, students who have studied “graphing” are under the impression that the only system in which graphs can be drawn is a Cartesian, or rectangular, system, each point being named by its distance from two perpendicular axes.
Nondecimal Slide Rules- and Their Use in Modular Arithmetic
Authors: Sowell Katye Oliver , Mcguffey Jon Phillip
Published: November 2020

Addition and subtraction are quickly done by means of two number lines placed one above the other.
Vectors in the Eighth Grade
Authors: Grant Nicholas
Published: November 2020

Vectors in the lower grades? Why not? When we arrive in a strange city and ask directions for going from our present spot to a desired place, we are usually told how to get there in terms of direction and distance. That is what a vector is, a direction and a distance. This article suggests how vector ideas can be introduced at the end of a first-year algebra course.
The Ways to Build a Box
Authors: Gilbert Edgar N.
Published: November 2020

SUPPOSE a carpenter wants to build a box (rectangular parallelepiped) out of boards all having the same nonzero thickness. The simplest constructions use six rectangular boards, one for each face. At each joint between a pair of boards, one of the boards covers the end of the other (fancier constructions using dovetailed or beveled joints will not be considered). Thus there is a choice of two ways to build each of the twelve edges. If these choices were independent, there would be 212 ways to build a box. Actually the number is smaller because each choice at an edge restricts choices at others.
Using the Laboratory Approach to Relate Physical and Abstract Geometry
Authors: Whitman Nancy C.
Published: November 2020

A Major difficulty of students of high hool geometry is in realizing and apreciating the difference and interrelaonsbip between conclusions reached by ductive procedures, such as measuring, and those reached by deductive procedures. Part of this difficulty tern from are fact that geometric fact and principles are accepted as true on the basis of aductive technique throughout the students' K-8 geometry program. Then suddenly, in high school geometry, students re told that fact and principle can be accepted only on the basi of deductive reasoning. To help students overcome this difficulty, the following laboratory lesson as presented.
Number Definitions
Authors: Brumfiel Charles
Published: November 2020

As badly as we handle our definitions, dictionaries do worse. Thi is some consolation.
Mathematics and Modular Scheduling—A Mod Medium?!
Authors: Thomas King D.
Published: November 2020

Marshall McLuhan, sage of the communication age, several years ago uttered his now renowned postulate: “The medium is the message.” More recently, be and his disciples in the science of media have “pun-ipu1ated” that message to “The medium is the massage” (1964, 1967). McLuhan's claim is that we pay too little attention to processes of experience and the vehicles of communication, with overemphasis on the content.
The Amazing Chiu Chang Suan Shu
Authors: Swetz Frank
Published: November 2020

In the mid-nineteenth century, China was rudely awakened from her two thousand year old xenophobic slumber by the ironclads and cannons of foreign visitors. Suddenly, the survival of the Middle Kingdom was no longer contingent on the moral ethics of Confucian humanism but rather depended on acquiring technological proficiency. Emissaries of reform pleaded their case before the Dowager Empress. Their memorials1 urged educational innovations as a basis for industrialization. Principally, they requested that the languages and sciences of the “foreign devils” be studied in order to decipher the secrets of the West's technological prowess. In the study of science, priority was given to mathematics as it was felt that this subject was basic to all other sciences.
Visual Aids for Relating Direct and Inverse Circular Functions
Authors: Viertel William K.
Published: November 2020

A visual aid for vividly demonstrating the relationship of the direct and inverse circular functions can be made quite easily and cheaply out of transparent sheets. The principle is simple, and the aid is similar to the graphs that are shown in all trigonometry texts. A graph of a trigonometric function is drawn on transparent material for an interval that makes it a one-to-one function. This graph is then turned over and rotated 90°, so that the x- and y-axes are interchanged and properly directed. The viewer now sees the graph of the inverse function with the proper domain and range.
Teaching Matrices Via Networks
Authors: Branfield John R.
Published: November 2020

MATRICES are now being used in a variety of situations—geometry, probability, geograpby, biology, statistics, costing, networks. The power of the matrix methods lies in the conciseness with which the matrix describes certain aspects of a situation and the techniques used to find out more about those aspects. No mathematical education is now complete without some knowledge and use of matrix methods. Of all the many possible situations in which to teach matrices, networks contain the teaching ingredients of simplicity at the beginning, intrigue and motivation, realistic problems (given and self-made), and sound development of the theory. The processes used in this article are not new; they have been put together as a suggested approach to the teaching of matrices.
New Products
Authors: Hansen Viggo P. , Johnson Randall E. , Koetke Walter J.
Published: November 2020

With many more teachers asking for ways to create instructional systems with an individual touch through multimedia presentations, a filmstrip-making kit would seem to be helpful. An article in the May 1972 issue of the MATHEMATICS TEACHER, “Individualizing Instruction with Homemade Filmstrips and Audiotapes,” set forth the details in designing a filmstrip.
Another “Zip the Strip”
Authors: Pedersen Jean J.
Published: November 2020

IN Clifford A. Long's article, “Zip the Strip,” in the January 1971 MATHEMATICS TEACHER, he indicated modifications of the Möbius-strip demonstration—that is, “cutting the strip along a curve one-third of the way across the strip and giving the original strip more than half a twist hefore cutting it down the middle.”
A Look at Regular and Semiregular Polyhedra
Authors: Stengel Carol Elizabeth
Published: November 2020

A polyhedron is regular if and only if each face is a regular polygon and each face is congruent, to every other face of the solid. The polyhedral angles of the solid must he congruent, as must the dihedral angles.
The Element of Surprise: An Effective Classroom Technique
Authors: Johnson David R.
Published: November 2020

To come upon suddenly and unexpectedly; to discover suddenly; to strike with a sudden feeling of wonder thai arrests the thoughts, as something unexpected or extraordinary. These are a few definitions of the word surprise found in a standard dictionary. Read these definitions again! Is the element of surprise present in your classroom NOW? I believe the element of surprise belongs there! In fact, I feel that the element of surprise can be one of the major teaching techniques used daily by mathematics educators. But wait a minute! What is “surprising” about arithmetic, algebra, or geometry? After all, isn't mathematics dead, dull, and difficult?
An Experiment in Teaching the History of Mathematics
Authors: Dittrich Alan B.
Published: November 2020

We expect high school students to ask why a particular manipulation is taught, what is the value of a whole area of mathematics, or what, benefit will follow technical adeptness, If we answer only that one skill is acquired to increase facility for another—as trigonometric identities make the solution of certain integrals possible, and truth tables aid in probability—then we may misrepresent and degrade mathematics and hone the unfortunate conviction that, beyond percentages, mathematics is the annoying plaything of teachers, the etherealization of irrelevance.
But What if it Doesn't Factor?
Authors: Collister Larew M. , McCabe Thomas R.
Published: November 2020

A young student teacher was presenting a lesson in the solving of quadratic equations in an elementary algebra class. In the introductory exercises every quadratic expression factored into two binomials with integral coefficients; the student teacher proceeded to set each factor equal to zero and solve for values of the variable.
Some Suggestions for an Informal discovery Unit on Plane Convex Sets
Authors: Olson Alton T.
Published: November 2020

This article is written to draw attention to plane convex sets as a possible supplemental unit in a high school ge-ometry course. The study of convex sets is udvisable for a number of reasons: (1) it has close ties with other branches of mathemtics, (2) it is geometrically intuitive in its initial stapes, and (3) the apparatus necessary for work in the field is not complex, that is, no analytic apparatus is used.
Mathematical Models and Applications: Suggestions for the High School Classroom
Authors: Mizrahi Abe , Sullivan Michael
Published: November 2020

In this paper wo support the position expressed in Max Bell's article in the April 1971 Mathematics Teacher, The point of view is that more attention to applications has been long overdue and severely neglected by most mathematics curricula, particularly those at the high school level. Bell points out that the reasons for this neglect are many, but he also emphasizes these reasons are not prohibitive and should not be deciding factors. In fact, more applications, more usable models, are needed not only in high school classes but also in elementary and college level classes. In this spirit, we wish to bring attention to additional relevant and timely applications of mathematics that can be implemented in certain high school settings.
A Note On The Teaching Of Algebraic Properties
Authors: Oxenrider C. J. , Hartung P. G.
Published: November 2020

Do your students really understand the basic algebraic properties of identity, commutativity, associativity, and inverses? Many teachers and textbooks ask their students to consider only familiar sets of numbers, such as the integers, with their usual addition and multiplication. When faced with the challenge of illustrating the more abstract nature of these algebraic properties by considering less familiar Structures, the tendency is to consider the student’s background too limited to make such examples feasible.
Fruitful Mathematics
Authors: Ranucci Ernest R.
Published: November 2020

I’M WRITING this article during a one-semester sabbatical leave in Quito, Ecuador. Quite impressive are the altitude (9,350 feet), our proximity to the equator (15 miles), and the friendliness of the Quiteños. Also worthy of note are the displays of fruits, vegetables, candies, and other goodies. Some are piled in the form of cones. Some are in the form of pyramids. Some are built on a rectangular base. I got curious as to the underlying principles behind these attractive displays and came up with these observations.
Are Triangles That Have the Same Area and the Same Perimeter Congruent?
Authors: Prielipp Robert W.
Published: November 2020

How would you answer this question: If two triangles have the same area and the same perimeter, are these triangles congruent? You might find it interesting to pause in your reading at this point and attempt to determine what the answer to this inquiry should be.
The Computer's Impact on Mathematics: Numerical and Monte Carlo Methods
Authors: Flynn Michael J.
Published: November 2020

Interesting computer methods for computing a definite integral are presented. High school or junior college students, as well as teachers of mathematics, should find this article worthwhile.
Two Preference Paradoxes
Authors: Usiskin Zalman
Published: November 2020

Even under quite reasonable schemes for decision making some sets of preference choices may not obey transitivity. The specific examples of such sets given here give rise to paradoxes because we so strongly believe transitivity should hold.
Programming Pascal's Triangle
Authors: Curley Walter
Published: November 2020

Computer programming can be used to extend a textbook topic.
Stock Exchange
Authors: Silverman Jerry Stuart
Published: November 2020

A stimulating game that provides some introductory ideas on stocks as well as opportunities for the maintenance of arithmetic skills.
Prime Numbers—a Locust's View
Authors: Sconyers James M.
Published: November 2020

Why is there a seventeen-year locust? In trying to answer this question, some people have been tempted to conclude that it's purely a matter of arithmetic. Can a case be made for bugs that calculate? Let's look at one explanation.
Clubs: Hubbard High School Mathematics Contest
Authors: Baker Betty L.
Published: November 2020

Since 1973 one of the most popular activities of the Hubbard High School (Chicago) chapter of Mu Alpha Theta bas been its annual Elementary School Mathematics Contest. The competition, which involves eighth-grade students from neighboring public and parochial elementary schools, is patterned after a once-popular Chicago area television quiz show called “It's Academic.” In 1980 sixteen schools and forty-eight students, selected by their classroom teachers, participated in this contest.
Transportation—a Rich Source of “Story Problems”
Authors: Riley James E. , Meyer Ruth Ann
Published: November 2020

Use some arithmetic and common sense to plan a bus route through some local towns.
A Geometric Interpretation of Series
Authors: Beynen John Van , McGinty Robert L.
Published: November 2020

At a conference in 1970 Edward Begle remarked that “our 'new math' experience of the 1960s has taught us a great deal about how to teach better mathematics, but very little about how to teach mathematics better” (1976). What follows is an example of a lesson on series that can be interesting and rewarding for your class.
Spherical Geodesics
Authors: Jamski William D.
Published: November 2020

What is the shortest distance between two points on a globe?
Applications: Camera Calculations
Authors: Fawcett George
Published: November 2020

On most single lens reflex (SLR) cameras there is a series of settings on a dial that regulates the amount of light reaching the film. The dial turns a mechanism called a diaphragm, which is a group of thin plates that vary the size of the aperture (the opening for light to enter the camera).
Computerized Testing: A New System of Evaluation
Authors: Noble John W.
Published: November 2020

An excellent teacher is often a demanding one. But implicit in the demanding is usually a heavy workload of evaluation that is time-consuming and generally menial in nature. There is a real need for teachers to be able to construct tests in a minimum amount of time to meet specific requirements such as testing time, content coverage, and level of difficulty. Once the instrument is developed, there is a desire to get test results quickly without the usual ordeal of grading. These needs can be met with a system called computerized testing, which consists of a computer, a computer terminal, and an automatic test-scoring device. A teacher employing some flexibility and resourcefulness can use computerized testing's enormous time-saving benefits to do a more effective and efficient job of instruction.
Applications: Estimating the Size of Wildlife Populations
Authors: Knill George , Fawcett George
Published: November 2020

Estimating is a method that can be used where it is impractical to count the actual number. For example, to determine the number of fish in a pond, one could drain the pond and count all the fish; however, the end result—dead fish—would not justify the means. The following is a better alternative.
Applications: The Mathematics of Sight
Authors: Knill George , Fawcett George
Published: November 2020

There is a distinction between the apparent size of an object and its actual size. Suppose you observe an airplane passing in the distance. If the airplane is far away, it may appear as a speck not much larger than a fly. On the other hand, if you observe the same airplane at close range—or even a fiy if it's on your own nose—you may be amazed at its size. Thus, the apparent size of an object depends primarily on the distance of this object from you. The actual size of an object is something that cannot be observed at all, because in order for our eyes to focus on the object it must be at some distance from the observer.
Other Numeral Systems—Alive and Thriving
Authors: Sizer Walter S.
Published: November 2020

The treatment of numeral systems other than our Western Hindu-Arabic one in most textbooks would suggest that other systems are only a relic of the past. In fact, many other systems arc currently in use.
Box Plots: Basic and Advanced
Authors: Mullenex James L.
Published: November 2020

Box plots are used for the purpose of analyzing and displaying important features of sets of data. More specifically, box plots are used as graphical representations of five-number summaries. Box plots and five-number summaries are new statistical techniques that were developed by John W. Tukey of Bell Telephone Laboratories. They are parts of a larger set of modern statistical techniques known collectively as exploratory data analysis, or EDA.
Soundoff: Are Teachers Prepared to Implement The Standards?
Authors: Fiske Michael B.
Published: November 2020

The National Council of Teachers of Mathematics's Curriculum and Evaluation Standards for School Mathematics (Standards) (1989) presents a view of precollege mathematics t hat stresses the development of mathematical power. Framed within the context of students' needs, societal expectations, and engaging teaching, the Standards proposes to define the mathematical content of school mathematics. It responds to the crisis in mathematics education described in Everybody Counts: A Report to the Nation on the Future of Mathematics Education (National Research Council 1989), A Nation at Risk (National Commission on Excellence in Education 1983), and Educating Americans for the 21st Century (National Science Board Commission on Precollege Education in Mathematics, Science, and Technology 1983). Although representing a consensus of mathematics educators, the Standards does not present a research basis for its recommendations (NCTM 1988) and thus at times stands at odds with the descriptive accounts of current mathematics teaching practices found in The Underachieving Curriculum (McKnight et al. 1987) and The Mathematics Report Card (Dossey, Mullis, Lindquist, and Chambers 1988). This article examines implications for teaching of explicit and implicit assumptions in the Standards and compares them with other views in the literature.
Implementing the Standards: Building key Concepts for Calculus in Grades 7–12
Authors: Jockusch Elizabeth A. , Mcloughlin Patrick J.
Published: November 2020

The NCTM'S Curriculum and Evaluation Standards for School Mathematics (Standards) (1989) recommends that the mathematics curriculum should include informal explorations of calculus concepts. These concepts can be developed as natural extensions of topics that students have already encountered. The approach should focus on exploring concrete problems in a way designed to build conceptual understanding of key ideas and to offer an introduction to some interesting applications.
Sharing Teaching Ideas: Collaborative Writing of Mathematics Problems
Authors: Carton Kitty
Published: November 2020

The NCTM's Curriculum and Evaluation Standards for School Mathematics (Standards) (1989) calls for opportunities for students to use mathematics as a tool for the communication of ideas. In this project, students in any level of mathematics, working in a cooperative, active setting, can develop their understanding of mathematical concepts through the collaborative writing of word problems. In so doing, they see mathematics from the inside out, as creators rather than mimickers; they are “doers” of mathematics, reflecting on and clarifying their own thinking about mathematical ideas in specific situations. Additionally, projects of the type described here can give teachers valuable information on which they can base further instructional decisions regarding the development of students' ability to communicate effectively using the language of mathematics.
Implementing The Standards: Students' Microcomputer-Aided Exploration in Geometry
Authors: Chazan Daniel
Published: November 2020

Four important themes presented in the K–12 Curriculum and Evaluation Standards for School Mathematics (Standards) (NCTM 1989) are mathematics as problem solving, mathematics as communication, mathematics as reasoning, and mathematical connections. The high school component also stresses mathematical structure. Furthermore, the Standards calls for new roles for teachers and students and suggests that microcomputer technology can help support teachers and students in taking on these new roles.
Is Nicolas Bourbaki Alive?
Authors: Walton Karen Doyle
Published: November 2020

Among the mysteries of mathematics, none is more elusive than the existence or individuality of Nicolas Bourbaki. Unlike most unsolved problems in mathematics, the resolution of the paradox cannot be found in a rigorous proof.
Sharing Teaching Ideas: Mathematics—An International Language
Authors: Denson Philinda Stern
Published: November 2020

“Mathematics—an International Language” is a large, colorful, and instructive bulletin board created by the combined efforts of the students in two geometry classes. It illustrates two facts: that the mathematics we study and use was—and is being—created by mathematicians around the world and that mathematics is a universal language.
Dependent or Inconsistent—Which is It?
Authors: Dorn Carl , Councilman Sam
Published: November 2020

When solving systems oflinear equations by Cramer's rule, students quickly learn that “if the determinant of the coefficients is zero, then the equations are either dependent (infinitely many solutions) or inconsistent (no solution).” The natural questions students should then ask are “What do I do next?” and “How can I tell which it is—dependent or inconsistent?”