bluuebunny/crossref_metadata_2025 huggingface dataset:
Topics of Interest to Mathematics Teachers: The Numerical Solution of Equations
Authors: Schaaf William L.
Published: November 2020

To obtain the solution of quadratic, cubic and quartic equations, a variety of algebraic methods are available. But in engineering and technological work it is frequently desirable to obtain approximate numerical values of all the roots not only of such equations, but also of higher degree polynomial equations as well as transcendental equations. Perhaps the best-known of several approximation methods is the scheme introduced by William George Horner in 1819, a method which is very efficient and relatively simple, although it entails considerable computation, and is limited to finding the real roots of a rational, integral algebraic equation. Inasmuch as Horner's method is quite familiar, it will not be described here.
Devices for a Mathematics Laboratory
Authors: Berger Emil J.
Published: November 2020

This section is being published as an avenue through which teachers of mathematics can share favorite learning aids. Readers are invited to send in descriptions and drawings of devices which they have found particularly helpful in their teaching experience. Send all communications concerning Devices for a Mathematics Laboratory to Emil J. Merger, Monroe High School, St. Paul, Minnesota.
Application
Authors: Myers Sheldon S.
Published: November 2020

We begin this first issue of 1951-52 with a renewed appeal for contributions. Those that you readers have submitted in the past have brought many favorable comments and have enabled Tub Mathematics Teacher, through its departments, to enrich the quality of mathematics instruction in this country. Perhaps this knowledge will encourage more of you to shift from passive to active beneficiaries of this journal.
Mathematical Miscellanea
Authors: Jones Phillip S.
Published: November 2020

A number of pupils in my trigonometry class asked whether it were not possible to prove the formulas for sin (x + y) and cos (x + y) by means of a simpler diagram than that given in the text book. In answer to that question, I developed the following proofs. Using one simple diagram, I found it possible to prove the formulas for the following:
On Relationships of the Affiliated Groups with the National Council
Authors: Mayor John R.
Published: November 2020

The plan for participation of the Affiliated Groups, which was used at the Twenty-ninth Annual Meeting in Pittsburgh in 1951, will be followed for the Thirtieth Annual Meeting to be held at Hotel Fort Des Moines, Des Moines, Iowa, April 16-19, 1952, with one addition.
Devices for A Mathematics Laboratory
Authors: Berger Emil J.
Published: November 2020

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

Approximately eighty students from eight high schools attended Eastern State High School's second annual mathematics conference for high school students on Monday afternoon and evening, November 19, 1951 at Eastern Illinois State College, Charleston, Illinois.
Devices for A Mathematics Laboratory
Authors: Berger Emil J.
Published: November 2020

This section is being published as an avenue through which teachers of mathematics can share favorite learning aids. Readers are invited to send in descriptions and drawings of devices which they have found particularly helpful in their teaching experience. Send all communications concerning Devices for a Mathematics Laboratory to Emil J. Berger, Monroe High School, St. Paul, Minnesota.
References for Mathematics Teachers: Teacher Education in Mathematics
Authors: Schaaf William L.
Published: November 2020

In the 1923 Report on the Reorganization of Mathematics in Secondary Education, it was pointed out that in some states the preparation of high school mathematics teachers was of such a low quality that the Committee judiciously refrained from giving the detailed data on those states. We have come a long way since then. Today, in general, secondary mathematics teachers are professionally and academically as well educated as teachers in any other field.
The factorgram
Authors: Swallow Kenneth P.
Published: November 2020

A simple proposition taken from the theory of numbers developed in such a way that the high school pupil cannot fail to be interested.
Notes from the Washington office: Activities of the State Representatives
Authors: Ahrendt M. H.
Published: November 2020

Almost from its beginning The National Council of Teachers of Mathematics has leaned upon a group of state representatives. The number of different state representatives who have served the Council during the thirty-four years of its existence is very great and our debt to them is incalculable.
Points And Viewpoints: National Council news
Authors: Wilcox Marie S.
Published: November 2020

A column of unofficial comment
Points and Viewpoints: What makes a good textbook review?
Authors: Rosskopf M. F.
Published: November 2020

A column of unofficial comment
Testing with a tangram
Authors: Mayer Haverly O.
Published: November 2020

Work with the tangram will prove to be different and a welcome break in class routine.
Mathematics in the Junior High School: Construct any angle with a steel tape
Authors: Eagle Edwin
Published: November 2020

The following, phrased in “to-the-student” language, describes an effective method for constructing angles of any size using only a steel tape. As presented here it would be appropriate for use with small groups of superior students at ninth-grade or even eighth-grade level.
Highlights of the Fifteenth Summer Meeting
Authors: Peak Philip
Published: November 2020

To all of you who were prevented from attending the Fifteenth Summer Meeting of the National Council held in Bloomington, Indiana, we offer our condolences. Many strange, interesting, enjoyable, and intellectual things happened at this meeting, and we can only bring you a brief word- and picture-summary of events.
Historically Speaking,— A historical puzzle
Authors: Court N. A.
Published: November 2020

Do the altitudes of a tetrahedron meet in a point? It would be unwise to accept, on a bet, the challenge of answering this question by “yes” or “no”: the answer would be incorrect in either case.
Questions with unusual answers
Authors: Bold Benjamin
Published: November 2020

Students like questions requiring them to demonstrate a knowledge of mathematical ideas. Here is one way to show students that conditions may not be satisfiable.
Historically Speaking,— Simon Stevin and the decimal fractions
Authors: Struik D. J.
Published: November 2020

Simon Stevin, the Flemish-Dutch engineer and mathematician, is best known in the history of mathematics as the inventor of the decimal fractions. This is substantially true, even if we acknowledge the fact that decimal fractions were used before Stevin.
Selection of topics from solid geometry for a one-year course in geometry
Authors: Small Dwaine E.
Published: November 2020

If solid geometry is to be taught with plane geometry rather than as a separate subject, what topics should be retained?
New Ideas for the Classroom: Decimals and duodecimals, .333 … or .400
Authors: McLean Robert C.
Published: November 2020

When the theory of numbers professor enters upon the discussion of never-ending decimals, he carefully points out that irrational numbers give non-repeating digits and that common fractions give repeating digits. A common example of this latter condition is one-third, well known as .33333….
Teaching load and qualifications of mathematics teachers
Authors: Brown Kenneth E.
Published: November 2020

Are mathematics teachers generally qualified to face the decisions of curriculum revision they must make? Do they have the time and facilities needed to make wise changes?
Tips for Beginners: An introduction to algebra with inequalities
Authors: Trine F. Dawson
Published: November 2020

Much can he gained in algebra by introducing inequalities with equalities. Equations and their solutions are more meaningfully taught in this way, and the student will get a clearer idea of the nature of a mathematical statement. By approaching equations and inequalities through the graphic approach, the student will discover methods of solving them.
Individualized algebra
Authors: Crosby Gwladys , Fremont Herbert
Published: November 2020

How can students learn to capacity without ability grouping?
Tips for Beginners: When I teach geometry
Authors: Chipman Hope H.
Published: November 2020

When I am teaching, especially in the junior high school and through the tenth grade, I try to give students the impression not only that mathematics is interesting, even exciting, but also that it is entirely possible to follow and understand it. I'll admit that in doing this I may not be giving a completely honest picture to some of the class members, but still, this is the viewpoint I try to put across.
Training of mathematics teachers in the USSR
Authors: Lindquist Clarence B.
Published: November 2020

This is a description of the education of today's mathematics teachers in the Soviet Union.
A new plan for instructing large classes in mathematics by television and films
Authors: Benner Charles P. , Rogers Curtis A.
Published: November 2020

How can television be used most effectively in teaching a course in mathematics?
Teaching of mathematics in Switzerland
Authors: Pólya G.
Published: November 2020

A description of programs, teacher education, performance, and change in mathematics teaching and learning in Switzerland.
Is solid geometry an entrance requirement for engineering schools?
Authors: Brant Vincent
Published: November 2020

A survey to discover whether engineering schools which require solid geometry for admission will accept a fused honors course instead.
Tips for Beginners: Pet peeves and pitfalls
Authors: Snelling W. Rodman
Published: November 2020

Some time ago a popular national magazine offered a regular feature column which developed a theme that can be paraphrased, “They ought to….” In that column readers noted their pet peeves and dared others to carry through plans to alleviate the situations described. In the same vein, I intend to point out my pet peeves in the teaching of mathematics with the hope that others may avoid the pitfalls that these small problems create in the classroom.
Junior high mathematics and the computer
Authors: Pierson Elliot
Published: November 2020

A teacher can take advantage of this approach to introduce new ideas in mathematics in a situation that stimulates interests.
A new qualification for mathematics teachers in the United Kingdom
Authors: Dodd W. A.
Published: November 2020

The Mathematical Association of England awards a “Diploma in Mathematics.”
Continental Classroom's TV course in probability and statistics
Authors: Mosteller Frederick
Published: November 2020

A report on experiences in teaching over a national television network
Annual high school mathematics contest
Authors: Salkind C. T.
Published: November 2020

Over 200,000 high school students are taking this national examination each year.
A property of the discriminant
Authors: Miller Vinnie H.
Published: November 2020

Invariance of the discriminant
Tips for Beginners: Graphical interpretation of the limit of an indeterminate function
Authors: Baylock Adrian
Published: November 2020

One of the first concepts encountered in the calculus, and surely the most fundamental, is that of a limit.
Using Historical Stories to Stimulate Interest in Mathematics
Authors: Harmeling Henry
Published: November 2020

Are we doing all we can to sell mathematics to our students? Do we enliven mathematics to make it a fascinating, stimulating science for them? What are we really doing to answer the ever-recurring question, “What good is the mathematics we are learning?” One of the best ways to stimulate curiosity and to instill a spark of the fascination of mathematics is to relate some of the absorbing true stories with which the history of mathematics abounds.
Tips for Beginners: A Christmas Graph
Authors: Holcomb John D.
Published: November 2020

An interesting Christmas picture emerges by drawing the following 23 graphs using the same axes. They combine graphs of circles, parts of parabolas, line segments and points. As can be noted, the graphs involve absolute value equations, as well.
Mathematics for the noncollege-bound in junior high school
Authors: Sassé Katharine J. S.
Published: November 2020

Junior high school is the time to keep doors open.
Problem solving in mathematics, I
Authors: Botts Truman
Published: November 2020

Good problems lead to further problems.
The end of a perfect number
Authors: Hanawalt Kenneth
Published: November 2020

A neat proof of an interesting property of perfect numbers
An illustration of the use of vector methods in geometry
Authors: Vaughan Herbert E.
Published: November 2020

Some interesting results which may help the teacher who will be teaching vector geometry
International Mathematical Education: The Mathematics Program in Japanese Secondary Schools
Authors: Fehr Howard F.
Published: November 2020

In Japan, education is compulsory for all children for the first nine school years, of which the first six years are elementary, and the last three years are the lower secondary school. Admission to the subsequent three-year upper secondary school is by examination, and usually 30 to 40 percent of the ninth-year students pass the examination and enter the upper secondary school. The branches of the upper secondary school are liberal arts, vocational training, and music and fine arts, all of which prepare students for university or higher study. This report is concerned with the mathematics curriculum at the lower secondary and the senior high school levels.
Mathematics and the Nobel Prize
Authors: Thomas à Kempis Sister M.
Published: November 2020

Recently it was my privilege to have access to the archives of Raymond Clair Archibald at Brown University. Among them I found his typewritten copy of a testimonial he gave honoring Edith Carlbourg, May 25, 1953. In preparation for this talk, he was urged to reminisce from his wealth of experience because of his travels as well as the nature of his work.
Further Evidences of Primeval Mathematics
Authors: Lloyd Daniel B.
Published: November 2020

In an earlier discussion on archaeological discoveries of a mathematical nature in Iraq [3],* the author described a 1958 digging operation near Baghdad. The present article describes another, more recent, digging in 1962 at Tel Dhibayi, a tell some four kilometers eat of the Tigris and within the area of one of Baghdad's new housing projects. The two diggings were a part of the same lraqi operation, but were separated by several years because of unfavorable climatic conditions of summer heat and winter rains.
Points And Viewpoints: A Time to Take Stock
Authors: Brune Irvin H.
Published: November 2020

A PROFESSIONAL journal thrives through the cooperation of many minds. As the calendar year draws to a close, we take stock of numerous debts to numerous people. We acknowledge our obligations herewith.
Friendly Numbers
Authors: Rolf Howard L.
Published: November 2020

Mystery, challenge, power, prophecy-these words describe the attitude men have held from ancient times concerning the natural numbers. While we would be reluctant to endow the natural numbers with mystical powers, we cannot deny the challenge they hold. They challenge us because many of the problem concerning integers are easily and simply stated, but the solutions have eluded man for centuries. In some cases the solutions have come to light slowly, and in other cases only in part.
Historically Speaking—: The Arithmetic of the Medieval Universities
Authors: Schrader Dorothy V.
Published: November 2020

The seven liberal arts—the trivium composed of grammar, logic, and rhetoric and the quadrivium made up of arithmetic, music, geometry, and astronomy— were the basis of the curricula in the medieval universities.
Area from a Triangular Point of View
Authors: Farrell Margaret A.
Published: November 2020

The creative mind balks at the ac cepted, the ordinary, the usual. Indeed, new mathematics seems to emerge as a reaction against the ordinary. If the role of the teacher is to encourage creativity by trying to guide students to think as mathematicians do, then accepted state ments must be challenged, alternate avenues explored, and comparisons made throughout the exploration.
Computer-Oriented Mathematics: Computer-Extended Instruction: An Example
Authors: Dorn William S.
Published: November 2020

In This article we will discuss the computer and its use as a laboratory in conjunction with a traditional mathe matics course. This type of computer use has a direct analogy with the use of a physics laboratory as it applies to a phys ics recitation class, We will make this analogy sharper in the next paragraph. Before doing so, however, we point out that the mathematics used in this article should be within the reach of most 8th or 9th grade students. That is, it uses only elementary algebra including some in equalities. However, the problems are interesting and difficult enough to challenge 12th grade students and, in fact, any pre-calculus student.
Experimental Programs: Prediction of Success in Junior College Mathematics
Authors: Morgan William P.
Published: November 2020

One of the most efficacious methods of objectively predicting student perfor-mance is through the utiliziaton of various standardized tests. It has been recognized for some time, however, that junior col-leges have not been given adequate con-sideration in the construction of standard-ized tests. That is, various tests have been developed for employment in high schools and traditional four-year colleges, but there are very few tests that have been designed specifically for the junior colleges. This problem can be partially solved through the construction of local norms for those standardized tests that are em ployed.
Enrichment or Acceleration?
Authors: Holmes Joseph E.
Published: November 2020

There has been much difference of opinion among secondary mathematics teachers during the last several decades as to just how the curriculum should be changed. There appears to have been, however, general agreement that changes were necessary. This agreement has given rise to the so-called accelerated programs in which students are enabled to study the usual (but modernized) high school courses at an earlier stage. In many cases this acceleration allows the student to be exposed to a full year of calculus while still in high school. Many of the student who have followed this route are unable to qualify for Advanced Placement credit and are required to repeat the first year of calculus as college freshmen. This indicates that we have allowed some students to enter an accelerated program who are not really ready.
Mathematics Educators Must Help Face the Environmental Pollution Challenge
Authors: Henderson George L. , Beck Mary Van
Published: November 2020

If Pollution of earth's air will cause heat from the sun to be trapped melting the polar ice caps and raising the level of the oceans 60 feet, how many people will either die or be left homeless?
Operations Research: A Lay Approach
Authors: Grinstein Louise S.
Published: November 2020

The want ads appearing in today's newspapers offer attractive salaries for people qualified in operations research. What is this field? Where is it used? What does it accomplish? Is it the panacea of modern man, providing all the answers to the complexities of today's world? How can students be readied for careers in this discipline? To dispel some of the aura of mystery attached to the words “operations research, (or OR, as it is commonly called), this article has been written.
Latin-Square Tiling
Authors: Gridgeman N. T.
Published: November 2020

Tiling a floor recently, I incorporated a six-color Latin square of quasi-systematic design and then began to ponder the number of variations that the pattern could be allowed without loss of its special characteristics. The question turned out to be interesting and illuminating.
Probability and the Platonic Solids
Authors: Kara Ryan Sister M.
Published: November 2020

The utilization of geometric models for probability experiments is not unusual. Each of the regular polyhedra Commonly known as Platonic solids (tetrahedron, cube, octahedron, dodecahedron, and icosahedron) can be used to generate a random sequence of numbers such as that normally generated with a set of ordinary dice. By identifying the faces of the polyhedra by a system based on Euler's Φ-function, a teacher can develop some interesting experiments for secondary school students.
The Key of e
Authors: Oliver Bernard M.
Published: November 2020

The first two topic were usually part of traditional curricula; the third often was. Complex numbers were first widely introduced in the “new” mathematics curricula, with their heavy emphasis on sets and numbers. There is good reason to question the validity of attempting to teach many of the abstraction contained in these new courses (especially when we do so with a rigor sometimes perilously close to rigor mortis) when, for the majority of youngsters, a better mastery of arithmetic or an introduction to calculu would be more widely use ful knowledge.
New Programs
Authors: Hight Donald W. , Kansky Robert J. , Richards Don K.
Published: November 2020

It is well known that some students are able to develop a high level of mathematical ability at a relatively early age. It is also a fact that many secondary schools are now sending to colleges and universities some students with excellent records of achievement, in high-quality mathematics programs.
An Analysis of a Simple Periodic Sequence
Authors: Palmaccio Richard J.
Published: November 2020

A recursive sequence is defined as a sequence of numbers in which each number is a function of previous numbers.
A Mathematical Vignette
Authors: Young Courtney D.
Published: November 2020

Mathematics educators should continually seek the opportunity to develop the following skills in their students: to abstract, to generalize, and to apply the concepts of mathematics. The cultivation of these fundamental tools can occur within the many content alternatives suggested for the traditional geometry course. Vector treatments, analytic treatments, and approaches through the study of transformations have been developed. It is our responsibility to become familiar with these various approaches. The experience gained now on a small scale will enable us to rationally choose the best combinations of synthetic, analytic, vector, and transformation methods as new materials become available.
A Note Concerning a Common Angle “Trisection”
Authors: Byrkit Donald R. , Waters William M.
Published: November 2020

THE trisection of an angle is a problem that frequently comes up in a beginning geometry course. An enormous amount of energy has been expended in attempts to “perform the impossible.” Many students find acceptable methods of angle triseetion by overlooking the requirement that it be done with compass and unmarked straightedge alone. The impossibility of this construction has long been established (Ringenberg 1968, pp. 245-250). It is possible, however, to make the construction if two marks can be made on the straightedge (Eves 1969, p. 85). Many other constructions are possible, some using higher plane curves, not to mention elaborately constructed mechanical devices (Yates, 1971).
The Forum: Accountability: Threat and Opportunity
Authors: Willoughby Stephen S.
Published: November 2020

Education has developed differently from other professions in our society. Probably because of its greater importance to society, education was socialized very early in history, while the other professions remained under the free enterprise system. You, as an individual, may believe that it is terribly important that you have excellent medical and legal aid when you need it, but to society it is more important that you have an excellent education. The legal and medical professions will attempt to help you whether your future contribution to society is likely to be positive or negative, but it is the goal of the education profession to help make your contribution to society more positive.
What?—A Roller With Corners?
Authors: Dossey John A.
Published: November 2020

Must wheels and rollers always be circular in shape? To answer this simple question, we need to study closed curves of constant width. First let us determine what types of curves might serve as the boundaries for the cross section of a roller. Then we shall consider which of these curves will define the shape of a wheel for a wheel-and-axle arrangement.
It Works for Base Ten and Other Bases Too
Authors: Minné Nels , Johnson Herbert N.
Published: November 2020

Current teaching practic spreads over a spectrum ranging from nearly impersonal dictation of information with lack of concern as to how students learn, to the encouragement of discovrry or invention of nearly all of the mathematics by the student.
Polyhedral Numbers
Authors: Hemmerly Howard
Published: November 2020

If a teacher is looking for a topic to suggest for the student with high interest and ability, he may find the answer in polyhedral numbers. This seems to be a fertile field for a student who has unusual curiosity and persistence. The pursuit of this study, though it takes time, should extend his capacity to visualize three-dimensional figures, to solve systems of linear equations, and to generalize results by the inductive process.
Some Probability Problems Concerning the Game of Bingo
Authors: Brown Gerald G , Rutemiller Herbert C.
Published: November 2020

We Have found the well-known game of bingo to be an excellent device for illustrating to students various laws of probability and a source for hypergeo-metric probability problems to supplement the familiar playing card examples. In particular, solution of the question “What is the expected length of a game of bingo as a function of the number of players?” provides students an opportunity to develop the expectation of a random variable theoretically and then to verify the result by computer simulation of the game.
A Method For Approximating The Value Of π with a Computer Application
Authors: Einhorn Erwin
Published: November 2020

The study of the transcendental number π is one that has fascinated mathematicians and students for many centuries. Archimedes, Ptolemy, Leibniz, William Brouncker, Ludolph van Ceulen, John Maehin, and countless others have suggested various methods for approximating the value of π. I should like to suggest a method that can be understood by good algebra students having the proper teacher guidance.
Cayley's Color Group
Authors: Moore Thomas E.
Published: November 2020

This article is presented in the hope of being the source of an enrichment topic for any class of capable students who have been introduced to the concept of a group. Thus we make the assumption that the reader understands the ideas of group and subgroup and possibly also of a normal subgroup and a set of elements that generates a group. Insight into the theory of groups can be derived, however, even if the reader is not acquainted with normal subgroups and generating sets.
The Nonparametric Approach in Elementary Statistics
Authors: Noether Gottfried E.
Published: November 2020

Introductory statistics courses are taken each year by hundreds of thousands of students across the country. These students come from many fields: the life sciences, humanities, education, agriculture, business, but above all from the social sciences. They rarely take statistics voluntarily. They sign up for the course because of departmental or graduation requirements. The great majority has minimal preparation in mathematics, rarely more than they bring along from high school. They carry over into statistics their prejudices of mathematics and quite often, justifiably so. Teachers of statistics courses should then ask themselves how they can make the introductory statistics course statistically meaningful and not simply an exercise in mathematics or, what may even be worse, a meaningless compendium of statistical techniques.
Some Implications of a2=|a|
Authors: Kepner James L.
Published: November 2020

This discussion of the handling and mishandling of a definition will be of interest to algebra teachers.
Numeration Systems with Unusual Bases
Authors: Ballew David
Published: November 2020

Teachers at nearly all levels find changing bases quite commonplace. The reader of this article is skillfully led up to the point where these new bases may be irrational or even complex.
Mathematics Applied in the Modern Bank
Authors: Boyer Lee E. , Hippensteel Philip J. , Lutz J. Robert
Published: November 2020

An example of mathematics being used to check advertisements about interest rates.
Why is geometry a basic skill?
Authors: Sherard Wade H.
Published: November 2020

Have you thought lately about why you teach geometry?
Clubs: Some Excellent Sources of Material for Mathematics Clubs
Authors: Lichtenberg Betty K.
Published: November 2020

Many of the articles published in the Mathematics Teacher are quite appropriate for direct use with high school mathematics clubs. The choices here are based on wide appeal, unusual themes, follow-up possibilities, and the number of accessible references, as well as other criteria that include personal preference. These ten articles are exciting, challenging, and highly suitable for student-presented programs and projects.
A Simple Proof from Elementary Algebra
Authors: Rizzuto James J.
Published: November 2020

“Prove it” is what students say when they don't necessarily believe something. For several reasons, this passionate skepticism does not appear in elementary algebra classes where “prove it” does not quite mean the same thing.
Clubs: Games and Projects
Authors: Allen Betty R.
Published: November 2020

Our mathematics club enjoys a game as part of each meeting. Favorite prizes are candy bars or other special treats.
In Defense of a Comprehensive View of Computer Literacy—A Reply to Luehrmann
Authors: Anderson Ronald E. , Klassen Daniel L. , Johnson David C.
Published: November 2020

In Arthur Luehrmann's critique of our February 1980 Mathematics Teacher article on computer literacy objectives, he proposes a rather narrow view of literacy. Despite his claim, there are two (not just one) generally accepted definitions of literacy. One is, as he points out, the ability to communicate, for example, reading and writing; and the other, which he neglects, is the state of being informed, “cultured,” and well versed. Whereas the first is a subset of the second, both definitions are commonly used. It is not surprising that the term computer literacy shares the semantic ambiguity of language literacy. The narrow view is that computer literacy is simply a matter of doing things with a computer. The comprehensive view is that computer literacy is an understanding of computers that enables one; to evaluate computer applications as well as to do things with them.
Applications: Skid Marks Estimate Speed
Authors: Knill George , Fawcett George
Published: November 2020

It is common knowledge that the faster a vehicle is going, the greater the distance required to stop after the brakes are applied. Conversely, the greater the braking distance, the faster the car must have been going when the brakes were applied. In the case of accidents, it is often important to estimate the speed from the braking distance.
Sharing Teaching Ideas: Seven Ways to Find The Area of a Trapezoid
Authors: Peterson Lucille Lohmeier , Saul Mark E.
Published: November 2020

We all want to get away from rote teaching, from students' memorizing and repeating to the teacher what the teacher has told the class. This struggle is particularly apparent in mathematics, in which much teaching is directed toward the mastery of formal tools of thought.
Sharing Teaching Ideas: The Legend of Soh Cah Toa
Authors: Camp Dane R.
Published: November 2020

When I first learned of the mnemonic SOH CAH TOA, I was told that a legend went along with it. Every year, when introducing trigonometric ratios in my precalculus classes, I searched to find such a story in an effort to anchor the acronym. I finally decided to create one myself. The result is the Trigonometric Review's exclusive story on the origin of SOH CAH TOA.
Student-generated Data in Elementary Statistics
Authors: Kundert Ken
Published: November 2020

The recent increased emphasis on the teaching of statistics in the high school classroom has focused primarily on the techniques of exploratory data analysis. Topics include stem-and-leaf plots, box plots, median-fit lines, and curve smoothing. A number ofhigh schools, however, still teach a course in statistics for the college-bound student. Included in this course are many of the classical topics of statistics generally found in an elementary statistics course taught to college students, with only intermediate algebra as a prerequisite. Although this article highlights selected topics in such a course and describes how student-generated data can be used to illustrate these topics, the basic idea can profitably be used throughout the mathematics curriculum.
The Baseball Schedule: A Modest Proposal
Authors: Kurtzke John F.
Published: November 2020

“How often does one baseball team play another team? What are the alternatives, and how could we figure them out?” Our local sportswriter can answer the first question–an American League team plays each team in its own division 13 times, and it plays each team in the other American League division 12 times; for a National League team, these numbers are 18 and 12, respectively. Yet if we were to ask our sportswriter about the alternatives, or how these would be determined, we would probably get a blank stare. Or the sportswriter might just say, “It's all done with computers”–the high-tech way of saying, “It's all done with mirrors.” In a sense, the sportswriter would be right–it is done with computers. Yet the challenge of it all is to understand what we do before we turn the problem over to the computer. So we shall see how to answer these questions and leave it to the Commissioner of Baseball to make any changes. The material here should be suitable for presentation at a mathematics club meeting, either by a student or by a teacher.
Procedures for, and Experiences in, Introducing Algebra in New South Wales
Authors: Pegg John , Redden Edward
Published: November 2020

It seems that children continue to have difficulty in understanding algebra. Some of this difficulty seems, to be due to problems in understanding algebraic concepts and algebraic symbols; some to the fact that children often do not seem to “reason algebraically” and perhaps most of all to the fact that children often do not understand what algebra is all about and what the purpose might be in studying it.
What I Wish I Had Learned in School
Authors: Tuttle Jerome E.
Published: November 2020

As an actuary, I earn my living by “doing mathematics.” Even though I'm not a teacher, I sometimes reflect on what I have and have not learned in my high school and college mathematics classes. I am lucky enough to have had many wonderful mathematics teachers throughout high school and college. Yet my teachers, constrained in their curricula in ways that I don't pretend to understand, never taught me a number of lessons that I have had to learn for myself.
Sharing Teaching Ideas: A Conceptual Approach to Solving Equations
Authors: Swain Stuart G.
Published: November 2020

I have developed a technique for solving equations that gives algebra students an intuitive grasp of this fundamental process while dispensing a healthy dose of graphing and computer experience
Chaos and Fractals
Authors: Barton Ray
Published: November 2020

Recently, Michael Barnsley and his colleagues at the Georgia Institute of Technology have been very successful in using affine transformations to create computer images of objects that occur in nature. The way the pictures develop on the computer monitor is very surprising and enjoyable to watch. I wondered if my students would be as surprised by, and interested in, these computer images as I was, so I introduced some of the basic ideas of Barnsley's work to them. I found that this topic generated an extraordinary amount of interest among the students.
Are the Primes Really Infinite?
Authors: Hudson Frank M.
Published: November 2020

Of course, the primes are infinite. This fact was first proved more than two thousand years ago. However, other facts might lead one to doubt this result. One such property of primes—their rarity—is developed in this article.
Ants, Tunnels, and Calculus: An Exercise in Mathematical Modeling
Authors: Winkel Brian J.
Published: November 2020

The process of mathematical modeling consists of a number of steps; these include accepting a situation, making some assumptions, converting the assumptions into mathematical statements, solving the mathematics, and interpreting the results in light of the situation.
Applications: Analyzing Data Relating to the Challenger Disaster
Authors: Tappin Linda
Published: November 2020

Memories of the Challenger disaster that occurred on 28 January 1986 are still vivid in the minds of many high school students. Thus, using data relating to this event can promote student involvement. This article introduces students to statistics by illustrating its vital role in decision making. Students at various grade levels with varying backgrounds will find this activity motivating and stimulating. Little or no background is necessary to appreciate this application of statistics involving exponential functions, mathematical modeling, probability, and curve sketching.
Problems, Patterns, and Recreations
Authors: Ajose Sunday A.
Published: November 2020

Apopular number problem asks solvers to make the integers from 0 to 20 using exactly four 4s.
Sharing Teaching Ideas: A Mathematical Investigation of Quality Control
Authors: Edgerton Richard T.
Published: November 2020

One way to apply the principles of the Curriculum and Evaluation Standards for School Mathematics (NCTM 1989) is to use real-world problems. The curriculum standards are enacted as students develop “mathematical power” while they communicate, reason, and make connections within and outside mathematics.
Inclusion of African American Students in Mathematics Classrooms: Issues of Style, Curriculum, and Expectations
Authors: Rowser Jacquelien Frazier , Koontz Trish Yourst
Published: November 2020

Recently we had the pleasure of observing high school mathematics classrooms in which teachers were demonstrating effective teaching strategies that they had learned at in-service sessions sponsored by the Eisenhower and National Science Foundation (NSF) programs. Unfortunately, we also noticed a distinct lack of African American students in the advanced mathematics classes. In most of the schools, Aftican American students made up 10 to 30 percent of the enrollment, but seldom was more than one African American student enrolled in the advanced mathematics classes.
Implementing the Assessment Standards for School Mathematics: Communicating Performance Criteria to Students through Technology
Authors: Lavigne Nancy C. , Lajoie Susanne P.
Published: November 2020

Mathematics education at all levels of schooling is currently undergoing change. Recommendations for improving the teaching, learning, and assessment of mathematics have been translated into standards that furnish guidelines for implementing change in mathematics classrooms (NCTM 1989, 1991, 1995). These standards emphasize the importance of engaging students in performance activities that require solving complex and realistic mathematics problems, reasoning about content and solutions, communicating understanding, and making connections among mathematical concepts.
Implementing the Professional Standards for Teaching Mathematics: External and Internal Characteristics of Learning Environments
Authors: Chapin Suzanne H. , Eastman Kristen E.
Published: November 2020

Mr. Dunn, a professional surveyor, has been invited to speak to a high school mathematics class.
Activities: The Inverse of a Function
Authors: Dyke Frances Van
Published: November 2020

Students have always found the topic of inverse functions difficult. When asked to find an inverse function algebraically, they often follow their first instinct, which is to take the reciprocal of whatever is given. This tendency is reinforced by the common use of the f-1(x) notation. If the function has taken an x to a y, we tell students that we want to take the y and find the process that will lead us back to the x. However, the idea of undoing what the function did gets lost in the mechanics of switching the x and y and then solving for the newly named y.
How High Is the Water Tower?
Authors: Alvarez Richard
Published: November 2020

When are we ever going to need this math?” This question, frequently heard in high school mathematics courses, elicits an interesting, and perhaps unique, response from teachers at one California high school. To experience an indepth practical application of mathematics, the trigonometry and precalculus classes meet at the school's water tower and use some of the mathematics learned in classrooms. Students measure the height of the tower's topmost point, which is not only inaccessible but also invisible from the ground. In the process, they engage in some good teamwork, see how their mathematics courses fit together to facilitate a task that no one course can do alone, observe how mathematics can work around physical obstacles, use their scientific calculators for chained and multistage calculations, and gain a little more appreciation for the physical facilities of their school. Students also see that mathematical analysis can be easier and more accurate than direct measurement or scale drawings. And every time they see the water tower―which is hard to miss—they remember that their mathematics really is good for something. As a special benefit, several interested students help prepare the activity and coach their peers.
Implementing the Assessment Standards for School Mathematics: Assessing Students' Thinking through Writing
Authors: Mayer Jennifer , Hillman Susan
Published: November 2020

Why do I ask my students to write? Why is writing important? What do teachers and students learn from the writing process? These questions cannot be ignored if the theme of communication in the Curriculum and Evaluation Standards for School Mathematics (NCTM 1989) and the guidelines suggested in the Assessment Standards for School Mathematics (NCTM 1995) are to be seriously considered.
Sharing Teaching Ideas: Max and Min Problems Using the Discriminant
Authors: Sarna Alan
Published: November 2020

The classic approach to optimization problems is currently accomplished through the use of a graphing calculator or, for advanced students, the techniques of differential calculus.
Journals: Pathways to Thinking in Second-Year Algebra
Authors: Chapman Kathleen P.
Published: November 2020

I was frustrated. It was another Monday morning, and my second-year-algebra students were seemingly half awake and definitely not “getting it.” What were they thinking? Were they thinking? I needed to find out. Since I wanted these students awake and aware, my mission was to find some means of discovering what they thought and how they thought. More important, I wanted my students to come alive in their thinking, not just to shrug and say, all too complacently, “Don't have a clue, Mrs. Chapman.”
Technology Tips: Using Interactive-Geometry Software for Right-Angle Trigonometry
Authors: Embse Charles Vonder , Engebretsen Arne
Published: November 2020

A student's first contact with right-angle trigonometry is usually a diagram like figure 1, showing a right triangle with the vertices labeled A, B, and C; the sides labeled appropliately a, b, and c; and the ratios using the adjacent side, opposite side, and hypotenuse displayed. Students compute the ratios for several, at most, numerical examples that show the trigonometric ratios for different angles. But what if students could actually explore this and other trigonometlic relationships in a straightforward, dynamic, and accurate way? Instead of doing a few static examples, what if students could investigate thousands of examples or observe a continuous-motion example showing trigonometric concepts more clearly? Interactive-geometry software like The Geometer's Sketchpad 3 (Jackiw 1995), Cabri Geometry II (Laborde and Bellemain 1994), or Tl-92 Geometry (1995) can be used to make these lessons “come alive” by allowing student to manipulate geometric constructions and discover mathematical relationships as the figure changes in real time.
Technology Reviews, Publications and Products
Authors: Flanders Jim
Published: November 2020

Prices on software, books, and materials are subject to change. Consult the suppliers for the current prices. The comments reflect the reuiewer's opinions and do not imply endorsement by the National Council of Teachers of Mathematics.
Technology Tips: A Mathematical Look at a Free Throw Using Technology
Authors: Embse Charles Vonder , Engebretsen Arne
Published: November 2020

Technology can be used to promote students' understanding of mathematical concepts and problem-solving techniques. Its use also permits students' mathematical explorations prior to their formal development in the mathematics curriculum and in ways that can capture students' curiosity, imagination, and interest. The NCTM's Curriculum and Evaluation Standards for School Mathematics (1989) recommends that “[i]n grades 9–12, the mathematics curriculum should include the refinement and extension of methods of mathematical problem solving so that all students can … apply the process of mathematical modeling to real-world problem situations” (p. 137). Students empowered with technology have the opportunity to model real-world phenomena and visualize relationships found in the model while gaining ownership in the learning process.
Mathematics and Psychology
Authors: Grove C. C.
Published: November 2020

Above we pointed out in rather homiletic form an attitude toward the attacks upon the disciplinary value of mathematics and indicated the proper reaction of teachers of mathematics to those attacks. Herein we proceed to sketch the attempts at “mental measurement,” to ascertain what was and is being measured, to investigate the fundamental assumptions made in the development of the statistical methods employed in the investigations on the transfer of mathematical training and to what classes of data we may apply methods involving the theory of probability. It is intended to give examples of classes of errors and misconceptions found in the literature, and it is hoped to be possible to make one or two helpful, constructive suggestions.
Sharing Teaching Ideas: Can You Guess My Number?
Authors: Maida Paula J.
Published: November 2020

One mathematical concept that students often find difficult is the role and use of inverse functions. In an effort to build students’ interest and confidence, I designed and implemented a worksheet and game in my precalculus class to introduce and illustrate the role of inverse functions.