bluuebunny/crossref_metadata_2025 huggingface dataset:
Main issues concerning the Soviet scientific, engineering, and educational challenge
Authors: Dewitt Nicholas
Published: November 2020

The author of Soviet Professional Manpower—Its Education, Training and Supply sets forth issues facing America—its citizenry and its teachers.
Modern mathematics in the twelfth grade
Authors: Kennedy Joe
Published: November 2020

The following is a description of an experimental twelfth-grade mathematics course taught by the author in 1955-1956 and 1956-1957 at Wisconsin High School, Madison, Wisconsin.
Graphical construction of a circle tangent to two given lines and a given circle
Authors: Mazkewitsch D.
Published: November 2020

Geometry teachers will enjoy this proof of a familiar construction problem.
New Ideas for the Classroom: Panel games
Authors: Johnson Donovan A.
Published: November 2020

A unique way to organize a classroom recitation period or to review a unit is to set up these activities as panel games.
Review tests can be different
Authors: Hazzard Louise
Published: November 2020

At the beginning of the school year we frequently give achievement tests and find that there are certain sections of the work of the previous year which the students have forgotten over the summer vacation.
Silken slippers and hobnailed boots
Authors: Jones Burton W.
Published: November 2020

The present situation in mathematics education has no parallel in the history of the schools. There is reason to be hopeful about future developments.
A punch-card “adding” machine your pupils can build
Authors: Collister Larew M.
Published: November 2020

An easily constructed device to use in teaching binary number notation and intersection of sets.
Some observations for mathematics teachers
Authors: Newsom Carroll V.
Published: November 2020

“We now have the opportunity, sought for so long, to exercise fundamental leadership in the creation of a better educational program in this country.” Some challenges and cautions relative to this leadership are presented here.
Measurement by telemeter or stadia
Authors: Rowles W.
Published: November 2020

A measuring device is described suitable for construction by the high school student.
Mathematical education and the scientific revolution
Authors: Cairns Stewart Scott
Published: November 2020

An appreciation of mathematics is a cultural necessity and command of its techniques a practical occupational need.
Mathematics and industry—a report
Authors: Wilcox Marie S.
Published: November 2020

The Committee on Co-operation with Industry of the National Council continues to seek to publicize instances of cooperation between industry and students and teachers of mathematics on the local level.
New thinking in mathematical education
Authors: Fehr Howard F.
Published: November 2020

Before reform can take place in mathematics education, reformers must consider the questions posed in this article.
New Ideas for the Classroom: New uses for the overhead projector
Authors: Hansen Viggo P.
Published: November 2020

One of the most effective visual aids for mathematics instruction is the overhead projector. It has great potential for adding meaning to the abstract symbols and processes of mathematics. It is a convenient, simple, and at the same time dramatic, device for illustrating and applying mathematical principles and operations.
A note on curve fitting
Authors: Santner Joseph F.
Published: November 2020

Writing a simple equation y = f(x) for an angle by using absolute value notation
Natural bases of logarithms
Authors: Jerison Meyer
Published: November 2020

Most Elementary Expositions of logarithms include the information that there are tables of logarithms to the base 10 and to the base e. The student who is told this finds the first bit of information perfectly understandable and the second perfectly mystifying.
Conic sections and their constructions
Authors: Burbach M. Annunciata
Published: November 2020

Conic sections are curves of great simplicity. Just as simplicity is indicative of great worth in human endeavors, these curves are of tremendous value to mathematicians, engineers, navigators, architects, astronomers, and physicists. The extensive usefulness of conic sections, which remained totally dormant for centuries, has been unfolded to us by scientists since the beginning of the seventeenth century.
Iterative methods in high school algebra
Authors: Hesse Allen R.
Published: November 2020

Iterative methods as enrichment
Historically Speaking,— The dawn of demonstrative geometry
Authors: Altshiller-Court Nathan
Published: November 2020

When the Greeks appeared in the history of culture, ca. 1000 B.C., Egypt, Babylon, and some other neighboring countries had already behind them more than a millennium of civilized existence. During that period of time, the Egyptians and the Babylonians succeeded in accumulating a considerable amount of learning, including mathematics.
A Survey Concerning Advanced Mathematics Curriculum
Authors: Blank William R.
Published: November 2020

What do chairmen of college mathematics departments and directors of admissions believe high schools should offer as advanced mathematics courses?
Projectiles
Authors: Stover Donald W.
Published: November 2020

Considerable facility with functions of angles in triangles is still demanded of students.
Experimental Programs: Science for a Day
Authors: Bissinger Barnard H.
Published: November 2020

To get at the nerve centers of future college students who might be interested in mathematics, and to give such students some idea of how they might use mathematics at a later stage in their careers, the Department of Mathematics at Lebanon Valley College, in cooperation with the Science Departments, tried an experiment which we found rewarding—indeed, a sheer delight. This is an account of that experience.
International Mathematical Education: Relations and functions
Authors: Steiner Hans-Georg
Published: November 2020

There arc certain types of binary relations which are of eminent importance for all parts of mathematics, including the elementary mathematics of secondary school. These are: order relations, equivalencerelations and functions. Pupils meet them at every stage of the curriculum.
New domains
Authors: Stone Edward J.
Published: November 2020

Enrichment without ennui
Additions to an annotated bibliography for careers in mathematics
Authors: Turner Nura D.
Published: November 2020

The November 1961 issue of the Mathematics Teacher carried an article entitled “A Bibliography for Careers in Mathematics” in which I spoke of the need to make career information available to both high school students and their teachers. The need still exists, and in an attempt to bring the annotated bibliography up to date I list further references that may be helpful to students and teachers.
Points and Viewpoints: Transition and appreciation
Authors: Brune Irvin H.
Published: November 2020

A column of unofficial comment
The Second Revolution in Mathematics
Authors: Allendoerfer Carl B.
Published: November 2020

“The first step in the learning of any mathematical subject is the development of intuition.”
International Mathematical Education: The development of the teaching of mathematics in Soviet schools
Authors: Cherkasov R. S.
Published: November 2020

When reading the article you will understand that the opening paragraph must be viewed against the background of the author's relation to his own government. The rest of the article is highly informative, enabling us to compare what has happened in Russia with what bas happened in the United States of America.—Howard F. Febr.
Experimental Programs: Research in the Teaching of High School Mathematics
Authors: Brown Kenneth E. , Abell Theodore L.
Published: November 2020

Since 1952 the U.S. Office of Education, in cooperation with the National Council of Teachers of Mathematics, has prepared a biannual summary of research in mathematics education. The present survey is based on the 645 replies from a thousand colleges that were sent the questionnaire. Many of the colleges responded even though they had no research to contribute to the study. In their answers they requested a report of the findings. About 125 of the reported studies for the calendar years 1961-62 were on mathematics education in grades 9-12.
The Cambridge Conference Report: Blueprint or Fantasy?
Authors: Adler Irving
Published: November 2020

In the title I ask two questions: Is the Cambridge Conference Report a blueprint giving detailed directions for the construction of the mathematics curriculum of the future? Or is it a fantasy that substitutes wishful thinking for an attempt to outline realistic goals that can be attained with flesh-and-blood pupils and teachers? I shall answer both questions immediately: It is neither. The rest of my discussion consists of elaboration and documentation of this answer.
Classics in Mathematics Education: Reaction vs. Radicalism in the Teaching of Mathematics
Authors: Myers G. W.
Published: November 2020

The writer did not choose his theme, but he very readily consents to discuss it, as the price of the privilege to appear on this program. There has not recently been a time in which it was more important to the teaching of mathematics than it is now to get right on mental attitudes towards reform in that teaching. The terms reactionary, conservative, radical, moribund, etc., are being bandied about, sometimes cast as aspersions upon otherwise altogether respectable folk, to such an extent that we need to know bow to comport ourselves gracefully when we are labelled for one of the groups. The appearance of such terms in periodical literature may be taken as a pretty sure indication that something in the nature of reform is operating on the minds of the users of the terms. It is readily admitted that no one need be disturbed whether his opponents classify him in the category to which he thinks he belongs; nevertheless, it is worth while to know what sort of impression one's attitudes, views, and arguments are making on those whose views he seeks to influence.
A Geometric Approach to the Conic Sections
Authors: Byrne Maurice Marie
Published: November 2020

Connie Byron, a student in an Algebra II class,* attempted a demonstration of how the locus of points fulfills the definition of a hyperbola by a geometric construction, based on the use of a circle as representing the constant difference. Although not conclusive in determining the locus at the time, this method was the stimulus for further study and investigation of a unique geometric approach to each of the conic sections.
Discovery-Type Investigation for Coordinate Geometry Students
Authors: Schaff Mary Ellen
Published: November 2020

Moving progressively downward through the curriculum from college course to advanced senior high school course to third-year high school course, analytic geometry has undergone surprisingly little alteration during the transition. At whatever level it is presented, it must incorporate some discovery-type investigation to maintain the interest of the better-than-average student.
Short Sequel to “Problem Solving In Mathematics”
Authors: Shoemaker Richard
Published: November 2020

One charming aspect of mathematics lies in the truism that the solution to a problem usually poses other equally intriguing problems, an example of which was carefully developed by Truman Botts in this journal for October and November 1965. Another serendipitous dividend in our discipline is the satisfaction derived from relating seemingly disjoint topics. The burden of this note is to observe the relationship existing between perfect numbers—that is, integers which are the sum of their proper divisors—and the problem taken from Volume II of the Hungarian Problem Book (this problem is to find three different positive integers such that the sum of their reciprocals is an integer).
Experimental Programs: Why Not a Survey Course in Mathematics?
Authors: Callow Wallace G. , Nichols Eugene D.
Published: November 2020

The twelfth-year mathematics’ offering remains an area of considerable discussion at council meetings and local curriculum studies. Most recent trends seem to be toward a broad spectrum of topics bearing little relation to one another. A rehash of the number systems, some set theory with some applications to probability, some linear algebra, and an elementary treatment of the derivative and definite integral often leave students with a false impression of what mathematics is all about. In short, it is only the tool value of mathematics which gains appreciation, in spite of the rigor we wave at this level.
On Some Geometric Inequalities
Authors: Klamkin Murray S.
Published: November 2020

In this paper, we derive some geometric inequalities (which are mostly well-known) in a simple manner by means of the Arithmetic-Geometric Mean Inequality.
A Reaction to “Definitions without Exceptions”
Authors: Rising Gerald R.
Published: November 2020

In the March 1965 issue of this journal, Charles H. d'Augustine expresses as his view that “it is better pedagogy to start with a basic definition which provides for the ‘fewest’ possible exceptions as a universe is ‘enlarged’.”1 He then offers as a case in point his recommendation for substitution of the crossproduct definition for multiplication for the definition of multiplication as repeated addition. This note is offered as a reaction to Mr. D'Augustine's continents, based on his own examples.
The Men Responsible for the Development of Vectors
Authors: Pawlikowski George J.
Published: November 2020

In Order to fully appreciate the study of vectors, one should have some idea of how they came about. This paper is devoted to the men responsible for the development of vectors. Since not all the men responsible for vector development could be discussed here, I have taken a few of the important men in the field and searched for information about them.
A Fortran Program for a Recursion Formula for Simultaneous Linear Equations
Authors: Simone Albert J.
Published: November 2020

A Previous issue1 presented a recursion formula for solving n linear equations in n variables. One of the major features of that formula is the ease with which it can be machine-programmed. While this feature was described, a computer program was not provided. A number of requests, however, have since been made for such a program. Moreover, an increasing number of mathematics departments now have an electronic digital computer capability. It therefore seems appropriate at this time to present a computer program for the formula.
Is Our Mathematics Inferior?
Authors: Carnett George S.
Published: November 2020

The United States has relatively fewer good mathematics students than many other countries.
Equations for Semicircles and Pings
Authors: Jolly Robert F.
Published: November 2020

In teaching the rudiments of graphing equations, one usually encounters the student's erroneous prejudice that only smooth curves can be described by elementary formulae; graphs such as bounded straight-line intervals or the union of several such intervals would not, he thinks, be the graph of a single equation, but rather would be described by families of inequalities. This prejudice seems to be encouraged by our present-day textbooks. In an effort to counter this tendency, one may attempt to challenge his students by raising the following question: How can you write a single equation the graph of which is your own name spelled out in block letters?
What is an Isosceles Trapezoid?
Authors: Ryoti Don E.
Published: November 2020

The November 1966 issue of The Mathematics Teacher contained an article by M. L. Keedy titled “What Is a Trapezoid?” In the article it is proposed that a trapezoid be defined as a quadrilateral with one or more pairs of parallel sides. A reason given is that when there is a choice between two definitions, the more general definition is preferred. Using this definition of trapezoid, a parallelogram is a special type of trapezoid. Thus, properties proven for trapezoids are automatically true for parallelograms. A difficulty with this definition, however, arises in connection with the isosceles trapezoid.
Conic Sections in Relation to Physics and Astronomy
Authors: Von Baravalle Hermann
Published: November 2020

Some of the earliest findings of math ematieal relationships are connected with conic sections. In 1594 Galileo discovered the laws of falling objects through his experimenta at the Leaning Tower of Pisa in Italy. The tower was particularly fit for his purposes as it is 54.5 meters in height with eight floors of balconies which go completely around the cylindrical structure. The tower leans over 4.3 meters from its vertical line. Objects dropped from various heights can be seen from the lower balconies, which serve as observation plat forms. The results that Galileo achieved were his two laws-first: the distances travelled by falling objects in successive time units are proportional to the odd numbers 1, 3, 5, 7, 9, …; and second: the distances travelled by falling objects and measured from the beginning of the fall are proportional to the square of the num-bers 1, 4, 9, 16, 25, …. Both relationships are shown in the diagram below (fig. 1).
Historically Speaking—: A History of Extraneous Solutions
Authors: Manning Kenneth R.
Published: November 2020

The assertion that before the rise of abstract algebra in the mid-nineteenth century, algebra was concerned more with the use of algorithms and rules than with investigations of logical foundations for the subject could well be tested by a de tailed study of the treatment of extraneous solutions throughout that period. A treat ment of extraneous solutions at a particu lar time in history would, because of the subtle mathematical principles involved in that treatment, reflect on the state of algebra at least as much as, and perhaps more than, any one topic in mathematics. However, the main purpose of this paper is to reveal the history of extraneous solutions, though reflections on the state of algebra seen from this historical account will be mentioned whenever appropriate.
Quadrarcs, St. Peter's, and the Colosseum
Authors: Gridgeman N. T.
Published: November 2020

1. Ovals. An oval is a regular closed curve in the plane, having at least one axis of symmetry and continuons curvature. However, if this last condition is re laxed, the definition will admit a class of ovals (in the everyday sense of the word— perhaps they should formally be called pseudoövals) that are important in applied mathematics. These are the composite closed curves made up of symmetrically disposed circular ares joined “normally,” that is, where tangent and normal are common, so that the abrupt change of eurvature is not apparent. We may regard such an oval as having an evolute that is not itself a curve but a set of isolated points-discrete centers of curvature. Ovals of this kind have long been familiar to architects and engineers. We shall now discuss the commonest type, the quadrarc, and afterwards look at two celebrated architectural examples.
An Old Stumbling Stone Revisited
Authors: Poole Robert R.
Published: November 2020

There have been several proofs ex hibited in the past for this classical geom etry problem. The following argument is, I hope, a significant improvement over earlier ones. The problem has been used traditionally as an example of an obvions proposition with an elusive proof.
A Case For Flexibility in Classroom Instruction
Authors: Genkins Elaine Kivy
Published: November 2020

Every conscientious teacher of mathe matics attempts to create the basis for good teaching by attaining as deep and broad a knowledge of mathematics as he can. On the immediately practical level, he plans his lessons in as much detail as he feels necessary to guide him in the teaching of each class. A creative teacher anticipates student inquiry and provides opportunity for student conjecture and exploration. However, inevitably there are times when a student asks a question or poses a conjecture which has not been anticipated and to which the answer is not immediately apparent. Over the years the writer has found that at these times it is beneficial in cases of genuine inquiry on the part of the students to depart from a planned lesson and follow up a student's question or conjecture with class discussion, thus capitalizing on the interest and motivation of the class.
Theorems For Finite Sets of Primes
Authors: Smart James R.
Published: November 2020

A Young mathematics student needs to formulate conjectures, to check their reasonableness, to state theorems of his own, and to verify them. In the early secondary grades, since he cannot always reason on the basis of a set of postulates, he sometimes needs to establish a theorem by simply cheeking all possible cases. Case-by-case inspection is, of course, only one method of proof, but it is one of importance for theorems about finite sets. The discovery and proof of such theorems as are in this paper should supplement work with general theorems capable of inductive proof.
An Intuitive Approach to Pierced Polygons
Authors: Jennings Donald E.
Published: November 2020

The approach described here was pro-posed after reading an article by Charles G. Moore, entitled “Pierced Polygons,” which appeared in The Mathematics Teacher, January 1968. The aim of that article was to introduce pierced polygons as an area of study that would lend itself well to experiment and discovery as a method of teaching.
The Junior High School Mathematics Program— Past, Present, and Future
Authors: Ferguson W. Eugene
Published: November 2020

Many teachers believe that the junior high school is “where the action is” in mathematics education today. Students at the junior high level usually are intellectually curious, full of life, and willing to spend some time exploring in depth mathematical ideas and concepts.
Computer-Oriented Mathematics: Polynomial Synthetic Division
Authors: Hoffman Irwin , Kauvar Larry
Published: November 2020

Synthetic division is a technique found in most high school algebra texts for dividing a polynomial by a linear term.
Try Graph Theory for a Change
Authors: Laible Jon M.
Published: November 2020

Do You need some fresh ideas to use in the classroom? Have you reached the point where both you and your students are tired of the day-by-day, topic-by-topic routine of mathematics presentation? Try graph theory for a change! Include a day or two of graph theory in a geometry course, in a senior mathematics course, or in any class you wish— just for something a little different. One of the beauties of graph theory is that little mathematics is necessary to begin its study.
Moment of Inertia Problem—A Classroom Paradox
Authors: Lehpamer Philip J.
Published: November 2020

I Recently assigned the following calculus problem to my class: Find the moment of inertia of a cone of radius r and altitude h about its axis, if its mass density is D. This problem caused considerable controversy among students and stimulated the entire class to discussion, for two different methods of solution had apparently produced two different answers.
Reviews and Evaluations
Authors: Eakin Richard R.
Published: November 2020

According to the authors, this text is designed “to meet the growing mathematical needs of future elementary school teachers.” The material impresses this reviewer as being suitable to meet that objective.
The High School Geometry Controversy: Is Transformation Geometry the Answer?
Authors: Gast Richard H.
Published: November 2020

Before I attempt to answer the question posed in the title of this article, I shall present the various proposals for and against the inclusion of transformations in secondary school geometry that have influenced my response.
A Reexamination of the Franklin Square
Authors: Jacobs Charles J.
Published: November 2020

In Chapter 3 of his book Magic Squares and Cubes W. S. Andrews states in concise form what is known about the particular magic square invented by Benjamin Franklin. Included in this chapter is “An Analysis of the Franklin Square” by Paul Carus. Each of these authors gives a most penetrating treatment of the subject, and an ingenious method of construction is described. Of this method, however, Carus admits: “We do not know the method employed by Franklin; we possess only the result, but it is not probable that he derived his square according to the scheme employed here” (p. 112).
New Mathematics for a New College
Authors: Brown O. Robert
Published: November 2020

In September 1968 the Federal City College of Washington, D.C., opened its doors to its first freshman class. There were fifteen hundred students. Federal City College had, and has, an open admissions policy; it is the first and the only public institution of higher education serving the residents of our nation's capital city. Since by definition a resident of the District of Columbia fails to reside in any one of the fifty states, he fails to have access to any in the impressive array of these states' public institutions of higher education unless he can pay premium tuition as well as meet selective admissions criteria.
From Graph to Formula
Authors: Assad Saleh
Published: November 2020

The modern mathematics curriculum places heavy emphasis on the notion of functional relations and on the development of graphing techniques. By the end of his junior year, or early in his senior year, the average high school student has little difficulty in identifying and sketching the graphs of linear, quadratic, and perhaps a few simple transcendental functions. His skill, however, is one-sided, for, given a set of data or a prepared graph, the same student is usually unable to discover the equation relating the variables.
New Products
Authors: Hansen Viggo P. , Johnson Randall E. , Koetke Walter J.
Published: November 2020

Polyominoes Kit (teaching kit) by TTT Mathematics Lab. Patrick Henry Junior High School, Granada Hills, Calif.$2.
Students Can Pose and Solve Original Problems
Authors: Arnold William R.
Published: November 2020

Many teachers would like to have their students discover relationships in mathematics but are not acquainted with a variety of means to accomplish such a goal. One approach used by teachers is to pose problems for students to solve that lead them to discover relationships that are characterized by conventional formulas. Certainly, junior high and high school students should do this.
Cooperation for Better Mathematics Teacher Education
Authors: Travers Kenneth J.
Published: November 2020

“CELIBACY,” the mathematicianphilosopher Alfred North Whitehead has observed, “does not suit a university. It must mate itself with action.” 1 In these action-packed days, people in the ivycovered halls constantly are being reminded of this maxim. And nowhere is it more nearly true than in teacher education. The long-overdue revolution in curriculum has made available instructional materials dealing with mathematical content relevant to the needs of our technological society.
On Squares in Arithmetic Progression
Authors: Luthar R. S.
Published: November 2020

MERTON Taylor Goodrich, in an article in the Mathematics Magazine (1966), developed a method to obtain square integers in arithmetic progression.
Criteria of Success in the Seventies
Authors: Adler Irving
Published: November 2020

The criteria of success; in the seventies may be different from the criteria of success in the sixties. We measure our success by the extent to which we achieve our objectives. While some of our objectins are the same in any period of time, there are some that depend on the time. To be able to formulate relevant and realizable goals for the next decade we have to ask ourselves, “What did we do well in the teaching reform movement of the sixties? What did we do poorly? And what did we leave undone?” We shall keep these questions in mind as we look back to the past and forward to the future while we consider separately fom major aspects of the movement to improve the teaching of mathematics: revising the curriculum, reforming methods, preparing teachers, and reaching pupils.
The Effect of the Use of Desk Calculators on Attitude and Achievement with Low-Achieving Ninth Graders
Authors: Cech Joseph P.
Published: November 2020

There appear to be two important reasons for the use of calculators with low-achieving students. One is to provide motivation, and the other is to improve achievement. The study discussed here addressed itself to the effect of the calcu-lator in these two areas.
True Theorems From False Conjectures
Authors: Manheim Jerome H.
Published: November 2020

In 1933 R. K. Morley noted that there are just four proper fractions with denominators less than 100 where illegitimate cancellation of a digit produces the correct answer.1
Some Convenient Fractions for Work with Repeating Decimals
Authors: Hilferty Margaret M.
Published: November 2020

Because my seventh-grade students greatly enjoy the work of expressing common fractions as repeating decimals, I have wanted a larger number of suitable fractions than is offered in the texts. My reasons are three: to allow for enrichment, to give opportunity for division drill in an experience that has proved pleasant, and to dispel the notion that repeating decimals are somehow magic numbers.
Ski Judge Bias
Authors: Bedford Crayton W.
Published: November 2020

Overheard at a recent ski meet after announcement of the results of the jump: “It's unbelievable! Anything in green gets two extra points from Baker.”
The Forum: Individualization of Instruction: How Much Individualization?
Authors: MacPherson Eric D.
Published: November 2020

If we attempt to identify the main causes of improvement in educational practice during this century, I believe that we must pay more attention to the development of general new sensitivities to children and the availability of new technical aids than to any deliberate applications of either educational philosophy or of educational research. Furthermore, the amus-ing (in retrospect) attempts to generalize the results of short-term one-factor laboratory experiments and normative surveys to the classroom disport many of us to applaud rather than fault these two bases for reform.
Some Whimsical Geometry
Authors: Pedersen Jean J.
Published: November 2020

This article deals with a method of approximating, by folding a paper strip, any regular polygon whose number of sides is of the form 2n + 1, where n is any natural number; or 2n+1 - 1, where n is any natural number greater than 1.
x=−b±b2−3ac3a
Authors: Mandelbaum Joseph
Published: November 2020

Almost everyone knows the quadratic formula. Indeed, we have learned it so thoroughly and so often that the first reaction of many people on looking at the title of this article will he, “That's wrong. It should be ….” We seldom consider, or encourage our students to consider, the possibility of solving other types of problems that have two solutions or other formulas that have “discriminants” like that of the quadratic formula. Perhaps there are other interesting formulas, similar to the quadratic formula, from which important mathematical concepts can be learned.
Discovering A Congruence Theorem: A Project Of A Geometry-For-Teachers Class
Authors: Smith Malcolm
Published: November 2020

College teachers who work with prospective high school teachers have noted that their students tend to “teach as they have been taught.” Therefore, when one of my students brought a problem to my geometry-for-teachers class, I decided to make the solving of this problem a class project. The ensuing investiga- gation was one of the most interesting and rewarding made by my classes.
A New Statement of Purpose—a Letter from the Editors
Authors: Johnson Donovan A. , Brant Vincent , Egbers Elden B. , Nichols Eugene D. , Rosskopf Myron F.
Published: November 2020

Even as this issue of the Mathematics Teacher is different, we expect all issues of volume 66 to have a new aspect. There is a different color, a different format, but more important, a different emphasis, As mathematics teachers, the members of the Editorial Panel are aware of the need of NCTM memhers for help in meeting classroom problems. Our de-termination to meet this need is expressed by our new statement of the purpose of this journal, namely, “The Mathematics Teacher is devoted to the improvement of mathematics instruction in the junior high schools, senior high schools, two-year colleges, and teacher-education colleges.”
The Forum: The Computer in Secondary Mathematics: The Computer-a Facilitator in Management and Instruction
Authors: Gibb E. Glenadine , Atchison William F.
Published: November 2020

In the midst of the rapidly changing field of computer development, one of the problems facing mathematics teachers in secondary schools and mathematics educators in colleges and universities is the optimal role of the computer in secondary school mathematics classes and the accompanying needs in teacher education to prepare teachers to use the computer in their classes. In 1965 the National Council of Teachers of Mathematics Committee on Computer Oriented Mathematics outlined available options. Since that time, others (including Gleason 1968; Zoet 1969; Alpert and Bitzer 1970; Hansen 1970; Travers 1971; and Jerman 1972) have dealt with the persistent question “How should we use the computer in our schools?” Two general directions seem to have emerged: (1) instructional individ-ualization through computer-managed in-struction (CMI) and computer-assisted instruction (CAI) and (2) the use of the computer as a computational device and as a means of simulating concepts within the present curriculum.
Numeration Bases and Infinity
Authors: Berlinghoff William P.
Published: November 2020

Ever since Zeno's arrow made the Creek world stand still, continuous infinity has fascinated philosophers and mathematicians alike. Lute in the last century, Cantor and Dedekind established the correspondence between the points on a line and infinite decimal expansions of numbers, opening a Pandora's box of confrontations between intuition and rigor. These many paradoxes, most of them related to Cantor's revolutionary work, began a period of unprecedented interest and research in the logical foundations of mathematics. The purpose of this article is to indicate how one of the unusual examples discovered early in that period might be used to encourage students to examine and unify several apparently diverse mathematical topics generally encountered in high school.
A Student's Construction
Authors: Stover Donald W.
Published: November 2020

A Student of mine hit on a construction for a line parallel to a given line through a given point; his construction seems clearly superior to the customary approach in textbooks. Others may wish to share it with their students.
All Three-Digit Integers Lead To…
Authors: Trigg Charles W.
Published: November 2020

HERE is a repetitive procedure that pupils will find intriguing. It requires only care and accuracy in subtraction, for which it provides some well-disguised drill. However, for the alert student, it offers the opportunity to dig out many underlying principles.
Recreation: Marker Solitaire
Authors: Spaulding Raymond E.
Published: November 2020

MARKER solitaire is an ingenious variation on the popular pegboard puzzles. Once a single form of the puzzle has been solved, simple variation of the rules or of the homemade solitaire board generates new challenges for gaming strategy and mathematical analysis.
A Mathematical Model and Its “Unrealistic” Consequences
Authors: Backler Jordan C.
Published: November 2020

In using mathematics to describe a physical process, the physicist often assumes conditions that, though they seem idealized, allow him to gain useful insights from his mathematical model.
A Geometry Game
Authors: Caballero James B.
Published: November 2020

Imagine two geometry students discussing their homework on the telephone. One student is trying to describe to the other the figure he has drawn for a problem. These students can converse freely about the figure, but neither one can see the other’s paper. This situation provides the idea for a sometimes hilarious game that can be an instructive mathematical recreation.
A Problem in Probability
Authors: Rudd David
Published: November 2020

This discussion makes use of two well-known rules for computation of probabilities. The first is that the probability that several independent events will occur in sequence is the product of the probabilities of the events. And the second is that the probability of an event that can occur in several mutually exclusive ways is the sum of the probabilities of the individual ways.
The Fourth Dimension and beyond … with a Surprise Ending!
Authors: Henry Boyd
Published: November 2020

Patterns for familiar figures are extended to higher dimensions.
Learning Packages for Mathematics Instruction—Some Considerations
Authors: Krulik Stephen
Published: November 2020

The uses of individualized learning packages are discussed and suggestions given for teachers who wish to develop their own packs.
A Discovery in Linear Algebra
Authors: Nicolai Michael B.
Published: November 2020

An eighth-grade teacher and his class share the excitement of discovery.
Matrix Algorithms for Operations on Polynomials
Authors: Goff Gerald K. , McKellips Terral L.
Published: November 2020

The authors show how matrix algorithms can be used in introductory algebra as well as in more advanced courses.
A Note on Magic Squares
Authors: Williams Horace E.
Published: November 2020

Excellent enrichment material for students of algebra. The author puts the study of magic squares into a meaningful mathematical context.
A Comment on “Fruitful Mathematics”
Authors: Sachs J. M.
Published: November 2020

Another approach to the predictor functions.
Geometric Proofs of Algebraic Identities
Authors: Horak Virginia M. , Horak Willis J.
Published: November 2020

An example of combining history, algebra, and geometry.
A Group Whose Elements Are Functions
Authors: Lichtenberg Donovan R.
Published: November 2020

An algebraic group with a different flavor.
The Mathematical Aspects of a Lunar Shuttle Landing
Authors: Anderson Bill D. , Lamb John F.
Published: November 2020

As the lunar shuttle with its load of tourists began its descent from orbit for a landing at Tranquility Base, an old, familiar sight appeared to the shuttle pilots— “earthrise.” Just as it had appeared for the first time to the astronauts aboard Apollo 8 in December of 1968 so many years ago, it now appeared for the first time to many of the tourists. There were the usual oohs and aahs as they tried to comprehend what they were seeing. The spell was broken by the pilot's announcement that they would be landing in a few minutes and that passengers should fasten their seat belts for safety. Just as the computer readout showed “ALT. 500 ft, SPEED 50 fps, REMAINING FUEL 100 units,” the automatic landing system failed. The pilot disengaged the disabled system and took manual control. The amount of fuel to be burned each second must now be entered manually in order to bring the shuttle in for a safe landing. A maximum of ten units may be burned each second, and three units per second are needed to offset acceleration due to the lunar gravity. If x units per second are burned, there is a decrease in velocity v of x − 3 feet per second (fps).
Using Determinants and Computers to Recognize Dependent and Inconsistent Linear Systems
Authors: Conklin Kenneth R.
Published: November 2020

An appeal for some lively discovery work on a routine topic.
80 Proofs from Around the World
Authors: Trono Tony
Published: November 2020

The following theorem was proved in “Reader Reflections” (December 1980). Theorem. If two angle bisectors of a triangle are equal, the triangle is isosceles.
Sharing Teaching Ideas: Formulas And Sugar Cubes
Authors: Pudelka Pam
Published: November 2020

In general mathematics classes, students can often use formulas correctly when the desired quantity is specifically identified in the problem, for example, “Find the area of a rectangle if the length is six feet and the width is five feet.” These same students do not always understand the relationship between formulas and their applications in the real world. This article describes a one-day activity that has been successful in reviewing perimeter, area, and volume by relating them to a readily available manipulative.
Activities: From Graphs to Matrices
Authors: Glidden Peter Lochiel , Laing Robert A. , Channell Dwayne E.
Published: November 2020

Introduction: The NCTM's curriculum and evaluation standards call for topics from discrete mathematics to be included in the 9–12 curriculum so that all students can “represent problem situations using discrete structures such as finite graphs, matrices, sequences, and recurrence relations; [and] represent and analyze finite graphs using matrices …”(National Council of Teachers of Mathematics, Commission on Standards for School Mathematics 1989, 176). This activity is offered as an example of how matrices can be introduced informally from finite graphs and how finite graphs can be analyzed by examining their matrix representations. Because this introduction to matrices is concrete and requires only marginal computational proficiency, it makes matrices accessible to the majority of middle and secondary school students.
Examples of the Use of Statistics in Society
Authors: Hinders Duane C.
Published: November 2020

Disraeli said, “There are three kinds of lies: lies, damned lies, and statistics” (Huff 1954). Although teachers should not dwell on the misuses of statistics, we do our students a disservice if we fail to point out that people sometimes carefully use statistics to make or “prove” a point. A starting point for teachers wishing to help their students deal with the world described by Disraeli can be a discussion of unusual and dramatic examples of the use or misuse of statistics or probability in society. It is hoped that the examples given here will inspire readers to start looking for their own interesting examples.
Why Women Succeed in Mathematics
Authors: Fabricant Mona , Svitak Sylvia , Kenschaft Patricia Clark
Published: November 2020

Contemporary educational research has suggested several factors that increase women's success in mathematics, and historical investigations support the validity of these recent observations. Most women in the history of mathematics shared three characteristics: a supportive family background, early exposure to significant mathematics, and available female role models in mathematics.
The Big Picture
Authors: Burke Maurice J.
Published: November 2020

I nformal methods for discovering and demonstrating geometric principles are commonly used in mathematics classrooms. This article demonstrates an informal method that I have used successfully in workshops and mathematics classes for the past five years. It helps to show that spatial visualization and analogy can be useful informal tools. The article also recommends a cluster approach when studying propositions of informal geometry.
Implementing The Standards: Multiple Approaches to Geometry: Teaching Similarity
Authors: Senk Sharon L. , Hirschhorn Daniel B.
Published: November 2020

Geometry as a subject uniquely furnishes a language for describing our physical world. It also gives a way visually to represent concepts and relations in other branches of mathematics. Although debate might always ensue on whether geometry should be a full-year secondary school course, the importance of geometry throughout a student's mathematics education seems to have broad acceptance. Consequently, it is not surprising that in the Curriculum and Evaluation Standards for School Mathematics (NCTM 1989) we find an explicit standard on geometry for all levels K–4, 5–8, and 9–12. In fact, for grades 9–12, two standards on geometry are included—one focusing on a synthetic approach, the other on an algebraic approach.
Applications: Finding Clusters: An Application of The Distance Concept
Authors: Clason Robert
Published: November 2020

We hope to make the usefulness of mathematics apparent when we teach our students. Relevant and convincing applications can motivate students and stimulate interest in mathematical discussion and exploration. We need a good supply of applications to avoid leaving students with the notion that mathematics is always abstract and leads only to more abstract mathematics.
Discretely Deranged Squares
Authors: Eckhardt Paul O.
Published: November 2020

The waves of change in entry-level college mathematics should not go unnoticed by those of us who teach at the secondary school level. One such wave, discrete mathematics, is lapping on the shore of the calculus. Many of our students will elect or be required to take college courses that include discrete combinatoric concepts. The focus of this article is to examine a special set of permutations and some of their extensions.
Sharing Teaching Ideas: “Modeling” In The Statistics Classroom
Authors: Voss Sarah H.
Published: November 2020

When teaching a beginning statistics course two years ago at a small midwestern women's college, I encountered an enormous amount of anxiety and lack of confidence, both on the part of the students and within myself. To the students, this course had bad public relations and a reputation for difficulty that far exceeded the actual level of mathematical skills involved. On my part, I faced the challenge of overcoming students' antipathy and fear-a task that sometimes seemed overwhelming.
Integrating Statistics into a Course on Functions
Authors: Schultz James E. , Rubenstein Rheta N.
Published: November 2020

For more than a decade calls have been made for the inclusion of statistics as part of the secondary school curriculum. Although some responses have been to add units to existing courses or to create special self-contained courses, we believe that statistics should be integrated into the secondary curriculum in ways that complement the mathematical and statistical learnings (Rubenstein et al. 1988).
The Point-Slope Equation Revisited
Authors: McNally Ann , Bonsangue Martin V.
Published: November 2020

Students in elementary algebra are typically taught that the point-slope form of a linear equation can be used to write the standard form of the equation, given either a point on the line, and the slope of the line or two points on the line. The necessary a lgebraic simplification inherent in t his procedure, although worthwhile for reviewing skills, can obscure the meaning of the standard equation of a line and its relationship to points on the line. This article presents an alternate approach that may be helpful in clar ifying both the mechanics and the meaning of deriving the standard form.
Applications: Two Geometry Applications
Authors: Boal Jan List
Published: November 2020

Fitting circles and straight lines together smoothly has been a standard problem in geometry since the time of Euclid. Last summer while working as a consultant for a machine-design company that manufactures precision components for projectile textile looms, I was asked to solve the two nonstandard geometry problems that follow. The problems arose in the design of a shuttle returner for a loom and required the construction of two circles that were to interface smoothly with another circle and two straight lines. In figure 1 is shown the end of the shuttle returner with the desired circles indicated at A and B.
A MathCAD Exploration: Hunting for Hidden Roots
Authors: Levy Benjamin N.
Published: November 2020

With the help of newer calculators and computers, we may now realistically ask more mathematics from our students. Not only should they be able to extend familiar mathematical concepts and tools to solve larger classes of problems, but they should also develop a deeper understanding of what it means for an equation to have a solution.
Integrating Writing and Cooperative Learning in the Mathematics Class
Authors: Artzt Alice F.
Published: November 2020

As teachers have recognized the importance of giving students opportunities to communicate their ideas about mathematics, they have begun to implement writing activities and cooperative-learning strategies in their classrooms (e.g., Abel and Abel [1988]; Artzt [1979]; Havens [1989); LeGere [1991); Mcintosh [1991]; Nahrgang and Petersen [1986); Sutton [1992]).