bluuebunny/crossref_metadata_2025 huggingface dataset:
Mathematics, the artistic science
Authors: Rosenbaum R. A.
Published: January 2021

“What can he more satisfying, more exhilarating, than to perceive an elegant mathematical solution of a significant practical problem? Nothing—nothing can surpass this—hut one experience can equal it: to find an unsuspected practical application of an elegant mathematical system.”
Tips for Beginners: Christmas at Palm Beach High School— “the geome tree”
Authors: Chancy Josephine M.
Published: January 2021

The pupils in the advanced geometry class of Palm Beach High School, West Palm Beach, Florida, decorated the main hall of their school building for the Christmas season. The decorations of the very attractive “Geome Tree” were made by students. The members of this class had received their introduction to the fundamentals of solid geometry by the time they decorated their unique “Geome Tree” with ornamental solid figures.
A statistical game and sample size
Authors: Glasser Gerald J.
Published: January 2021

Students of mathematics have often learned about mathematics and some of its applications from studying various games of skill and chance.
A community of consultants
Authors: Levy Norton
Published: January 2021

How can teachers get help from mathematicians and scientists who live in the community?
Historically Speaking,— The New York Fragments of the Rhind Mathematical Papyrus
Authors: Guggenbuhl Laura
Published: January 2021

The Rhind Mathematical Papyrus, a document which needs no introduction to the student of the history of mathematics, was purchased in 1858 in Luxor by the English Egyptologist, Sir A. Henry Rhind.
Tips for Beginners: An Impromptu Discovery Lesson in Algebra
Authors: Snyder Henry D.
Published: January 2021

In a study of quadratic equations and quadratic functions, my second-year algebra class embarked on a discovery lesson which turned out to be the most successful class session I have ever had. The lesson plan called for a consideration of quadratic inequalities and their solutions, but we began as usual by discussing the homework of the previous day.
A Magic Square for the New Year
Authors: Gardner Martin
Published: January 2021

How does one construct a magic square of this amazing type?
Positive and Negative Factors in Team Teaching
Authors: Korb Sister Mary Victor
Published: January 2021

ALTHOUGH team teaching appears to have received its initial impetus in the English and social studies curriculum areas, it has much to offer in mathematics education, also. While team teaching varies from school to school, as it does from department to department in the same school, the entire program is dependent upon the faculty, curriculum, and physical facilities as well as upon administrative approval and cooperation.
The Use of Elastic for Illustrating Homothetic figures
Authors: Arcache Alexander
Published: January 2021

A FIGURE F' is said to be homothetic to a figure F with respect to a fixed point I called center of homothecy, and a constant ratio k, called ratio of homothecy, if there is a one-to-one correspondence between the points of these two figures such that:
Have You Read?
Authors: Peak Philip
Published: January 2021

It is only rarely that one should report on the report of a report, but this time it seems sufficiently special to warrant the extra action.
A Review of Discovery
Authors: Bittinger Marvin L.
Published: January 2021

Learning by discovery is the result of a rebellion against the authoritarian, lecture, or tell-to-do method of teaching. In this article learning by discovery is described as any learning situation in which the learner completes a learning task without extensive help from the teacher. In discovery learning the teacher's role may vary from careful guidance to no guidance at all.
Classics in Mathematics Education: Mathematics in the History of Civilization
Authors: Court Nathan Altshiller
Published: January 2021

The discussion of the subject at hand would gain in clarity if I could ask the members of my invisible audience whether they can remember themselves far enough back in their childhood, when they were not able to count—that is, when they could not recite the series of words one, two, three, four, and so on. Radio being what it is, the question must remain unanswered. Fortunately, I know what the answer would be, if it could be given. It is just like asking somebody whether he can recall the time when he could not walk or when he could not talk.
Achievement and Programmed Learning
Authors: Sneider Sister Mary Joetta
Published: January 2021

With the continual developments in programed instruction, educators are responding in varied ways. Advocates assure that programmed instruction provides for the individual differences of students. Goldberg and his colleagues grant the advantages of programmed learning to be most evident with respect to the slower learner in helping him to obtain a direct familiarity with course materials.1 Fillmer finds that it guarantees personal attention to each student working with it.2 Clark claims, “A well-built program will allow a pupil to learn mathematics, to learn good mathematics, and to learn it with interest and understanding.”3 Heimer holds that there are three values to the student who is using programmed instruction: (1) He is continuously involved in the learning process, (2) he gains immediate reinforcement, and (3) he proceeds at his own rate.4
The Relationship Between Nontrivial Automorphisms and Order of Fields
Authors: Keiser Victor
Published: January 2021

In the article “A Nontrivial Automorphism of the Field of Real Numbers” (THE MATHEMATICS TEACHER, December 1966), Robert F. Lawler defines operations * and Δ on a certain set F which he refers to as the field of real numbers. Before going further, let us point out that the use of the definite article in the phrase “the field of real numbers” is justified by the well-known theorem stating that any two complete ordered fields are isomorphic; it does not arise from the existence of some particular distinguished set of objects which we call the real numbers.
In the Name of Geometry
Authors: Hillman Thomas P. , Sirois Barbara
Published: January 2021

A Very popular parlor game for teenagers and adults, one which centers upon the identification of famous international personalities, is called “In The Name Of.…” The joy of the game results from its requirement to create and decipher abominable puns, as illustrated by the following example. “Mrs. Ensey thought the movers had taken all of her personal possessions from the house when suddenly she discovered a large empty wastebasket in one of the closets. She called her Chinese houseboy and said, ‘The movers must take this wastebasket with them.’ The houseboy sighed and said, ‘Madame too late!’ ‘Why?’ asked Mrs. Ensey, and the houseboy replied with the name of a very famous Dutch painter.” (The answer is Van Gogh, of course. Did you get, it?)
Good Triumphs over Evil
Authors: Prielipp Robert W.
Published: January 2021

One day while strolling through the park Mr. Good encountered Mr. Evil. After the exchange of a few brief remarks concerning the weather, Evil invited Good to accompany him to the ice cream parlor for a strawberry sundae. Immediately after Good had accepted his invitation, Evil challenged Good to a contest. Evil suggested that they determine who would pay for the sundaes by flipping two pennies—one to be provided by Good and the other by Evil.
The Five-pointed Star
Authors: Boyer Lee E.
Published: January 2021

The construction technique for inscribing a five-pointed star in a circle is practically equivalent to that for inscribing a pentagon, which is fairly well known. The proof involves a rather striking combination of geometric, algebraic, and trigonometric thought. For this reason the topic of the five-pointed star can be an excellent enrichment exercise for eleventh- or twelfth-grade mathematics students.
Ninth-Grade Algebra, Programmed Instruction, and Sex Differences: An Experiment
Authors: Easterday Kenneth , Easterday Helen
Published: January 2021

This experiment was originally designed to evaluate the use of programmed instructional materials in ninth-grade algebra. A thorough analysis of the data from the experiment reveals some interesting, and perhaps valuable, differences when the sex of the students is considered. Does the sex of the students have any effect on their success with programmed instruction?
Theoretical and Practical Solutions for a Geometric Sequence
Authors: Lyng Merwin J.
Published: January 2021

During a recent discussion of the sum of the terms of a geometric sequence, I posed the following problem to my class.
A Plausibility Argument for L'Hôpital's Rule
Authors: Ash Carol
Published: January 2021

Here is something to think about!
Classics in Mathematics Education: Some of Euclid's Algebra
Authors: Evans George W.
Published: January 2021

If Euclid had called his Elements the Elements of Geometry, that name would have been a misfit, for it was geometry only to a minor extent. It included the elements of all the mathematical knowledge of his time, as well as of much in times far distant. Many things had been separately known, the achievements of different wise individuals following paths apparently diverging. He was able to see the unity in these scattered treasures and had the skill to frame a system embodying that unity. He found it in number.
The Heart of Teaching
Authors: Trimble Harold C.
Published: January 2021

Think about some good mathematics teachers you have known. For the moment, don't try to define your usage of the word “good” in a precise way. Perhaps it means “effective.” But, then, what is an effective mathematics teacher? You may conclude, as I do, that individual teachers of mathematics may be effcctive in either of two quite different ways.
Have You Read …?
Authors: Peak Philip
Published: January 2021

This article is a summary report of the Conference on Needed Research in Mathematics Education held at Athens, Georgia, under the sponsorship of three groups, including the National Council of Teachers of Mathematics. However, this entire issue of the Journal is devoted to the conference, and you will probably want to read it all.1
In-Service Education in Mathematics: Self-Service In-Service
Authors: Bye Nikoline
Published: January 2021

SEVERAL years ago I had a cry for help from a former student. This student, after graduation, had taught mathematics for several years, then married and had a family. Now the children were in school; and, feeling that she could be away from home, she decided to return to teaching. However, during the interval that she had been busy raising a family, the material taught in mathematics classes, as well as the method of teaching it, had changed considerably. “Now,” she wrote me, “what do I do?”
Reviews and Evaluations
Authors: Eakin Richard R.
Published: January 2021

This book was written for those who have missed, for some reason or other, a good background in the fundamentals of arithmetic. It is a clear-cut exposition which expects a certain mature attention. Concepts are presented logically in an easy-to-follow narrative style, with mathemetical terms written in italics the first time they are presented.
Letter to the Editor
Authors: Hockett C. F. , Augulis Vida
Published: January 2021

In his interesting article in The Mathematics Teacher in the April 1968 issue, pages 304-95, Earl K McGeehee, Jr., fails to mention one nomenclatural point of considerable importance: A Hausdorff space that is also a door space is a house door space.
The Division and Euclidean Algorithms in the Gaussian Integers
Authors: Jordan James H.
Published: January 2021

In elementary courses of algebra and number theory one runs across the well-known division algorithm.
A Hunch and a Proof
Authors: Verno C. Ralph
Published: January 2021

In an introductory course for freshmen not majoring in mathematics, I assigned the simple combinatorial problem of finding the number of committees of five that could be formed from a group of fifteen people.
International Mathematical Education: The Education of Mathematics Teachers in Other Countries
Authors: Fehr Howard F.
Published: January 2021

To Give an entire account of the teacher preparation in all countries would indeed be a lengthy process, the writing of which would require several volumes.
On the Shape of Plane Curves
Authors: Dotson W. G.
Published: January 2021

The intuitive but somewhat vague idea of the shape of a curve can be formulated precisely in an elementary way that would seem to have a natural place in plane analytic geometry. The formulation of such a definition and the systematic investigation of the resulting mathematical structure should constitute a valuable and interesting exercise for students in analytic geometry courses or combined calculus-analytic geometry courses. However, the topic does not seem to have been included in any of the standard texts.
What Points Are Equidistant from Two Skew Lines?
Authors: Forsythe Alexandra
Published: January 2021

One spring my twelfth-grade analytic geometry students were restless. They were surfeited with the bite-sized pieces of theory parceled out in their text. What they needed was an interesting problem whose solution would require them to organize and fit together some of the facts they had been learning. This required a fairly diligent search, but we finally located the necessary problem. The statement of this problem has the double advantage of being brief and easy to understand. I found the problem intriguing myself. Here it is: Find the points equidistant from two skew lines. (Two lines in space are skew if they do not lie in a single plane.)
Reviews and Evaluations
Authors: Eakin Richard R.
Published: January 2021

Discovering Modern Algebra, as the name implies, has for its emphasis discovery rather than exploration in depth. As a matter of fact, the author in his Preface says (in part), “As an undergraduate I always treated with suspicion a mathematics book containing the word ‘elementary’ in its title. To one of my addled wits this invariably meant pain, anguish, and considerable hard labor. By that level this is a subelementary text. It deals with certain aspects of ‘modern mathematics’ in a lighthearted and tolerably earthy fashion and is meant as a leadin to more formal texts.”
In-Service Education in Mathematics: Student Mathematics Achievement as Related to Teacher In-Service Work
Authors: Whitmore Edward H. , Norris Fletcher R.
Published: January 2021

The need for adequate programs of continual growth for teachers has long been espoused. Several theories have been advocated about just what should be included in in-service programs, as well as about their duration and organization.
A Classroom Illustration of a Nonintuitive Probability
Authors: Kleber Richard S.
Published: January 2021

Kemeny, Snell, and Thompson1 give as an example of a nonintuitive probability the “matching-birthdays problem.” Briefly stated, the problem is to determine how many people there need be in a room in order that the probability is at least one-half that there are at least two people in the room with the Same birthday. The answer, assuming 365 equally likely birthdays, is much lower than most people would expect.2
Friday-the-Thirteenth
Authors: Bailey William T.
Published: January 2021

Superstitious or not, the problem of how many times a Friday-the-Thirteenth appears in a given year has the following elegant solution.
Have You Read …?
Authors: Peak Philip
Published: January 2021

Here is a two-page article each of your students and colleagues should read.
Computer-Oriented Mathematics: Prime Triplets
Authors: Koetke Walter , Hirsch Joseph
Published: January 2021

It is generally agreed that good teaching of mathematics encourages the student to create and investigate original problems. This article was motivated by a study of prime numbers, and illustrates how familiarity with a simple property of numbers can lead high school students to original research and hence to a genuine interest in mathematical theory.
Pretesting for the College Boards
Authors: Stover Donald W.
Published: January 2021

Before a newly written question can appear on the Level I or Level II Mathematics Achievement Tests administered by the College Entrance Examination Board, it must be pretested with a sample group of several hundred students. Such groups are carefully selected for their similarity to the populations taking the Level I or Level II examinations in previous years, in order to provide reliable indicators of the way the question will “behave” if used at a later date in College Board Examinations. Both before and after pretesting, the questions are scrutinized by the staff of the Mathematics Department at Educational Testing Service and by a Committee of Examiners consisting of seven teachers of mathematics at the secondary and college level.
Historically Speaking— the “New Mathematics” in Historical Perspective
Authors: Eves Howard , Wren F. Lynwood
Published: January 2021

There are probably very few teachers of mathematics today who have not been asked at least once the question: “What is the ‘new mathematics’?” In an effort to build an informative background on the evolution of mathematics as a serious study in our schools, one is almost compelled to review at least four periods of history. These four periods are (1) 300 B.C.; (2) the second quarter of the seventeenth century; (3) the second quarter of the nineteenth century; and (4) the four decades of the twentieth century from 1925 to 1965.
Sums of Consecutive Positive Integers
Authors: Prielipp Robert W. , Kuenzi Norbert J.
Published: January 2021

What is mathematics? For many secondary school pupils, it is solving equations, evaluating determinants, using synthetic division, or finding a derivative. All too frequently what the student does is more closely related to bookkeeping than to real mathematics. Perhaps this is why we have so many prospective mathematic majors at the university level who breeze through calculus but who never seem able to cope with “abstract mathematics.” Their manipulative skills are adequate, but their ability to think has never been developed.
Using Discovery in the Calculus Class
Authors: Shilgalis Thomas W.
Published: January 2021

It is often assorted that the mathematics student who has been led to discover a generalization is more likely to understand, appreciate, and retain it than the student to whom it is presented in an expository manner. If this assertion is true, even for a limited class of “readily discoverable” generalizations, it behooves the mathematics teacher to consider a discovery approach to a theorem that he wants his class to learn.
Recreation: Eight-Marker Puzzle
Authors: Spaulding Raymond E.
Published: January 2021

The eight-marker puzzle is a variation of the ancient game of checkers, but it is played on just one row of squares. Playing the game helps students develop problem-solving ability and gaming strategy analysis.
Partitions in Algebra
Authors: Jacobs Claire C.
Published: January 2021

One purpose of teaching mathematics is to provide students with abstract models that will help them to analyze the phenomena that surround them. The greater the number of situations to which a model applies, the more fundamental it is, and the more useful a part of the student’s repertoire. Therefore a concept that is essentially simple, that has a vast number of applications, and that serves, the psychologists indicate, as a model of a fundamental learning process, would seem to be a prime candidate for inclusion in the school mathematics curriculum. Such is the set theoretic concept of partition.
Experiments Leading to Figures of Maximum Area
Authors: Moulton J. Paul
Published: January 2021

A sequence of ingeniously simple lab-style investigations reveals deep principles relating area and perimeter with important applications.
The Circular Geoboard— A Promising Teaching Device
Authors: Hutcheson James W.
Published: January 2021

This article includes sample lessons that illustrate how to discover a relationship and then prove that it is correct.
The Sixteenth International Mathematical Olympiad and Some Implications
Authors: Greitzer Samuel L.
Published: January 2021

The American team placed second in our first participation in an international mathematical competition.
On Improving the Efficiency of the Rational Root Theorem
Authors: Liberi Albert W.
Published: January 2021

An approach to finding the rational roots of polynomial equations.
Junior High School Mathematics: Motivation vs. Monotony
Authors: Sobel Max A.
Published: January 2021

Ten specific suggestions for motivating junior high school students to learn mathematics.
Using Cubes for Factoring Cubes
Authors: Metz James
Published: January 2021

This geometric interpretation of factoring the sum and the difference of two cubes should prove interesting for algebra and geometry students.
I “Found Well”
Authors: Cunningham Helen
Published: January 2021

A meeting in a Moscow museum.
Detective Stories: An Aid for Mathematics and Reading
Authors: Crouse Richard , Bassett Denise
Published: January 2021

Whet the problem-solving appetites of your mathematics students with these excursions into the deductive thinking of mystery reading.
A New Look at the “Centers” of a Triangle
Authors: Lightner James E.
Published: January 2021

Here is an interesting approach to the problems associated with medians, altitudes, and angle bisectors in a triangle; teachers and students should find this an informative addition to the usual geometry course.
The Fourth U.S.A. Mathematics Olympiad
Authors: Greitzer Samuel L.
Published: January 2021

The U.S.A. Mathematical Olympiad, now in its fourth year, has, to judge from the correspondence we have received, attained an enviable position in the world of secondary school education. Recommendations from school officials on behalf of superior mathematics students increased markedly this year. Participation is by invitation only.
Mathematics Contests and Mathletes
Authors: Flener Frederick
Published: January 2021

What activities are available for the mathematically talented students in your school? Do you have a mathematics club to discuss unusual mathematical problems? Many states have mathematics contests, such as the Annual High School Mathematics Examination, in which a simple general test is given.
Isoscelesn
Authors: Ranucci Ernest R.
Published: January 2021

An isosceles triangle can be a source of many ideas. How many isosceles triangles ca11 one isosceles triangle give birtlr to? Read the article to find out.
Individualized Mathematics—A Simple Approach
Authors: Osborne Ursula
Published: January 2021

Senior high school mathematics teachers will be interested in reading how this teacher provided individualized instruction to students that were mostly of Latin-American cultural background.
A Discovery Approach to the Cube Root Algorithm
Authors: Bolduc Elroy J.
Published: January 2021

A logical explanation for the cube root algorithm through discovery.
The Subtle Scales of Justice
Authors: Lepowsky William L.
Published: January 2021

A descriptiue analysis of a balance scale in use
Discovering a Formula That Generates Even Perfect Numbers
Authors: Pasquali Giorgio
Published: January 2021

This article illustrates how an amateur mathematician found a formula to generate perfect numbers. This method can be used to suggest a creative assignment for high-ability students.
Dynamical Systems: Birkhoff and Smale
Authors: Kaplan James , Strauss Aaron
Published: January 2021

Beginning perhaps with Newton in the seventeenth century many brilliant thinkers have attempted to express the observable behavior of the physical world in the form of relatively simple mathematical formulas and equations. Frequently these equations are “differential equations.” which relate the position of some object (or objects) to its velocity, acceleration, temperature changes, energy, external forces, and so on.
Your Council at work
Authors: Glenadine Gibb E.
Published: January 2021

The Spirit of '76—a familiar expression this year as the United States celebrates 200 years as a nation. During these 200 years, Americans, from diverse backgrounds and experiences, have had the opportunity to fashion their own lives and the life of their society. Their Bicentennial is a recognition of two centuries of a new nation among nations—a nation described by some as one that has sought to weave a tapestry of different colors, beliefs, and ideas that is both resilient and brilliant. As with any weaving, flaws and mistakes are to be expected. Through these mistakes, Americans have learned the value and importance of debate, discussion, interaction, and involvement. During this Bicentennial year, individuals with common interests in civic, industrial, business, religious, and educational or other professional endeavors are taking advantage of the opportunity to renew their commitments and revitalize their efforts. For each of us, it is a time of reflection as well as rededication.
The Weather as a Markov Process
Authors: Hoffman Nathan
Published: January 2021

Activties with spinners lead students to a better understanding of chance processes as well as provide a simple model for the weather.
A Special Way to Remember July 4, 1776
Authors: Hiatt Arthur A. , Dichiera Debra A.
Published: January 2021

Some numerical drill and algebra wrapped into an intriguing exercise.
Games in Senior High School Mathematics Classes
Authors: Gilman Sister John Frances , Rowe Sister Joan , Hildenberger Sister Mary Frances
Published: January 2021

Carefully selected games can help students win at mathematics.
An Old Card Trick Revisited
Authors: Felps Barry C.
Published: January 2021

Here is an old trick explained and extended to decks of any number of cards your students may select. This can he an intriguing way to wrap up a unit on modular arithmetic.
“Zero”: An Addition Game With Integers
Authors: Miller G. Frank , Jeffrey Maurice
Published: January 2021

An easy-to-play card game that provides drill for junior high school students.
Completing The Cube
Authors: Turner Barbara
Published: January 2021

Classic summation formulas come to life with elegant geometric models.
Student Attitudes Toward An Audio-Tutorial Course in Precalculus
Authors: Mayes Vivienne , Brigham Lucille , McNeill Sarah Virginia
Published: January 2021

A report on a successful experiment to provide for differences in abilities, backgrounds, and career goals.
A Product of Our Times
Authors: Greenwood Jay
Published: January 2021

A fresh way to stimulate arithmetic practice and the exploration of number patterns.
The Orthotetrakaidekahedron—A Cell Model for Biology Classes
Authors: Wahl M. Stoessel
Published: January 2021

An interesting geometric model has some surprising applications.
Operative and Nonoperative Group Definitions
Authors: Buchthal David C.
Published: January 2021

A group is a group is a group. However, group axioms are arbitrary. What happens if they are altered?
The Artist as Mathematician
Authors: Slawsky Norman
Published: January 2021

Mathematical ideas evolve in many and varied ways. Although some are consciously reflected in works of art, others find their origin there.
An Investigation of Integral 60° and 120° Triangles
Authors: Muller Richard C.
Published: January 2021

Here is an example of how to use a computer to follow up your conjectures in geometry and trigonometry.
Anyone can Trisect an Angle
Authors: Reyerson Hardy C.
Published: January 2021

Two interesting curves from analytic geometry come to life as “solutions” to a classic problem.
Activities: Polyhedra from Cardboard and Elastics
Authors: Woolaver John
Published: January 2021

For years teachers and students have labored to construct models of regular and semiregular polyhedra using cardboard and glue. It is considerably easier to eliminate the glue and substitute elastics [rubber bands]. All the Platonic and Archimedean solids can be quickly assembled and disassembled if sufficient cardboard faces and elastics are available.
A “weight” Watcher's Test
Authors: Hoffman Joseph R. , Henry Boyd
Published: January 2021

Do you weigh your mass or mass your weight? Let's find out.
Mathematics Projects in Junior High School
Authors: Bruckheimer Maxim , Hershkowitz Rina
Published: January 2021

A sequence of problems to promote creative problem-solving habits using basic tools of algebra 1.
Your Biorhythm and the Sine Curve
Authors: Kelterborn Bonnie
Published: January 2021

An instant application of sine curves to a topic guaranteed to provoke some interesting discussion in your class.
There is a Lot of Mathematics in Two Buckets
Authors: Goff Gerald K.
Published: January 2021

How many quantities can you create with two buckets?
Constructing the Parabola Without Calculus
Authors: Bruckheimer Maxim , Hershkowitz Rina
Published: January 2021

A new, simple, and insightful approach to an old problem.
On the Rationality of Exponentials and Logarithms
Authors: Williams Richard K.
Published: January 2021

When is ab rational? Irrational?
Poker Probabilities: A New Setting
Authors: Litwiller Bonnie H. , Duncan David R.
Published: January 2021

Discover some surprising outcomes when playing six-card poker.
Randomness in Physics and Mathematics
Authors: Troccolo Joseph A.
Published: January 2021

An “absurd” problem yields mathematical insights into atomic physics.
Teaching Quadratic Problem Solving
Authors: Stover Donald W.
Published: January 2021

A single powerful theorem to solve quadratic equalities and inequalities.
Mysticism and Magic in the Number Squares of Old China
Authors: Swetz Frank
Published: January 2021

Numerology and the use of minicalculators can be combined to provide fascinating classroom activities.
Line Reflections in the Complex Plane— a Billiards Player's Delight
Authors: Musser Gary L.
Published: January 2021

A target for your cue ball can he found using complex number computations.
Some Thoughts on “Minimal Competencies”
Authors: Forbes Jack E.
Published: January 2021

What is mathematical literacy? How do we achieve it for all students?
State Minimal Objectives and Testing, the Michigan Council's Experience
Authors: Clason Robert G.
Published: January 2021

This model for accountability was carefully developed. Needed modifications will continue to be made.
The College Board Scholastic Aptitude Test: An Overview of the Mathematical Portion
Authors: Braswell James S.
Published: January 2021

From time to time I have been asked to speak to groups of high school mathematics teachers about the mathematical portion of the Scholastic Aptitude Test (SAT). This article affords an opportunity to provide current information about this test to a greater audience of mathematics teachers and others interested in mathematics education.
Back-To-Basics in Mathematics: An Educational Fraud
Authors: Offner Carl D.
Published: January 2021

Some thought-provoking arguments that should cause you to think about your philosophy of mathematics.
Activities: Calculator Capers
Authors: Schmalz Rosemary
Published: January 2021

Materials: Calculators and one set of worksheets for each student.
Derivatives on the Hand-Held Calculator
Authors: Morgan Raymond V. , Warnock Tony T.
Published: January 2021

Are you depending on a convincing demonstration of simple limits using a pocket calculator? Caution is advised.
Efficient Algorithms for The Calculator
Authors: Friedlander Richard J.
Published: January 2021

Algorithmic design is often considered the one justification for computers in the classroom. Now this payoff comes with calculators.
A Current Event for The Mathematics Classroom
Authors: Watson Jane M.
Published: January 2021

Often there are not enough data available for a model.
Basic Competencies in Vermont
Authors: Kenney Robert
Published: January 2021

The Vermont Board of Education, at its meeting on 20 July 1976, accepted a set of basic competencies in reading, writing, speaking, listening, and mathematics. Action on competencies in reasoning was deferred for further development.
A Mathematics Club Project from Omar Khayyam
Authors: Lumpkin Beatrice
Published: January 2021

An ancient construction yields Solutions to cubic equations.
Approximate Integration: Comparative Examples
Authors: Venit Stewart M.
Published: January 2021

Some useful examples to contradict the idea that Simpson's rule is best.
X-Game—Some Consequences
Authors: Young Courtney D.
Published: January 2021

Play this game, enjoy it, analyze it, and then create your own version.
Number Theory for Secondary Schools?
Authors: Pagni David L.
Published: January 2021

Some interesting opportunities for reinforcing computational skills and using a minicalculator.
Extremal Problems on A Geoboard
Authors: Lott Johnny A. , Nguyen Hien Q.
Published: January 2021

There are many fascinating geoboard problems and patterns that are of interest to high school students. One of these is the nine-point circle, presented on an isometric grid geoboard by Robert L. Jones in the February 1976 issue of the Mathematics Teacher.
Applying The Technique of Archimedes to The “Birdcage” Problem
Authors: Stannard W. A.
Published: January 2021

A very difficult problem falls to an old but powerful method of analysis.
Who is Quickest with Mind and Fingers?
Authors: Kerr Steven D.
Published: January 2021

An enjoyable calculator activity.
Duplicating The Cube with A Mira
Authors: Martin George E.
Published: January 2021

Apollo, through his oracle at Delos, proclaimed to the Delians that if they were to be rid of the plague they must construct a cubical altar that would be double the size of the existing one. Because of this tale, the problem of the duplication of the cube has become known as the Delian problem. Legend has Eratosthenes reporting that the problem was sent to the geometers in Athens who were with Plato in the Academy. Plato responded that in setting the task, the god was shaming the Greeks for their neglect of mathematics and their contempt for geometry.
Measure for Measure
Authors: Trimble Harold
Published: January 2021

Several examples of ways algebra and geometry provide alternative views of a concept in mathematics.
Some Mathematics on The Balance Scale
Authors: Wright David G.
Published: January 2021

A common mathematical riddle involves a balance scale and a number of coins, which are to be sorted in a certain number of weighings.
Inscribed Polygons Lead to An Interesting Limit
Authors: Kich Kenneth
Published: January 2021

Recently, while working with polygons inscribed in a circle, I had an idea that led to a very interesting limit. The concept that got me started was the notion that the area of an inscribed n-sided polygon approaches the area of a circle as n approaches infinity.
Pattern Discovery with Binary Trees
Authors: Pauker Andy
Published: January 2021

These integers have parents. Try following their ancestry.
Another Look At √2
Authors: Lindstrom Peter A.
Published: January 2021

This sequel to the irrationality of √2 relies on notions of terminating and repealing decimals.
A Unification of Two Famous Theorems from Classical Geometry
Authors: Maor Eli
Published: January 2021

Two ostensively different geometry theorems are shown to be essentially the same if viewed from a new perspective.
Phantom Points
Authors: Duncan David R. , Litwiller Bonnie H.
Published: January 2021

Two common activities in advanced algebra concern the simultaneous solution of equations and the intersections of their graphs.
Bald Eagles, Sperm Whales, and Mathematics Teachers
Authors: Paul Clyde A.
Published: January 2021

A shortage of mathematics teachers now exists and will likely become more severe.
The Probability of Winning Dice Games
Authors: Litwiller Bonnie H. , Duncan David R.
Published: January 2021

One of the most popular dice games is Craps. The rules for a basic version of the game are as follows. Two dice are used. One player becomes the shooter for the first game. In subsequent games, other players take their turns as shooter.
On The Radii of Inscribed and Escribed Circles of Right Triangles
Authors: Hansen David W.
Published: January 2021

Mixing the Pythagorean theorem, the area of a triangle, and some first-year algebra yields some unexpected results.
Guessing, Mathematical Induction, and A Remarkable Fibonacci Result
Authors: Crawford John A. , Long Calvin T.
Published: January 2021

Anyone who has endeavored to teach mathematical induction is aware that this powerful method of proof is often only poorly understood by students and is frequently accepted as legitimate only after a certain amount of intimidation by the teacher.
Questions About Numbers Outside The Mathematics Classroom
Authors: Smart James R.
Published: January 2021

Test yourself and then test your students.
An Iterative Method for Computing Solutions to Equations Using A Calculator
Authors: Waits Bert K. , Schultz James E.
Published: January 2021

In this article we illustrate a powerful method for computing solutions to certain equations such as those involving higher degree polynomials, exponentials, and trigonometric functions. The method relies on basic algebra and the use of a scientific calculator. It is an interesting calculator application and serves as a review of important algebraic topics including functions and graphing.
A Neglected Minority
Authors: Goldman Myron W.
Published: January 2021

One alternative for helping inner-city students.
A Computer Meets a Classical Problem
Authors: Leonard William A. , Pagni David L.
Published: January 2021

Here we see not only how computer methods can be applied to old problems but also how computer efficiency affects the solution.
Synthetic Division for Nonlinear Factors
Authors: Reimann Kurt W.
Published: January 2021

Division of a polynomial of first or greater degree by another polynomial of first degree only is commonly performed by the process known as synthetic division.
Some Common Puzzles and Their Solutions
Authors: Bohland Bob
Published: January 2021

In teaching mathematics for the last fifteen years, 1 have used many puzzles. Sometimes they are used as part of a regular class routine, whereas at other times they provide a reward to those students who finish their work early and who want the challenge of solving mind-boggling problems. The following puzzles are a few that my students have found exciting and challenging.
What Should Not Be in The Algebra and Geometry Curricula of Average College-Bound Students?
Authors: Usiskin Zalman
Published: January 2021

Arguing against topics already in the curriculum—what this article tries to do—is certain to arouse emotions. Every topic in the curriculum is beloved by some mathematicians or mathematics teachers. If there were not such a constituency, how would the topic have become standard in the first place? We see today how difficult it is it to change practice. Every topic entered the curriculum at some time with what were, for that time, compelifng reasons for its inclusion.
Students’ Affective Responses to Mathematics: Secondary School Results from National Assessment
Authors: Carpenter Thomas P. , Corbitt Mary Kay , Kepner Henry S. , Lindquist Mary Montgomery , Reys Robert E.
Published: January 2021

Desired outcomes of instruction in school mathematics usually include some mention of achievement in mathematics and positive affective responses to the subject matter. Not only do we want our students to develop some expertise in mathematics, we are also interested in promoting their feelings of success and encouraging them to pursue their study of mathematics.
Application: Fingerprints and Fractions
Authors: Knill George
Published: January 2021

Since the beginnings of criminal investigation, police have sought an infallible means of human identification. The first method of identification was devised by a French police expert, Alphonse Bertillon, in 1883. The Bertillon system was based on a detailed description of the person, combined with full-length photographs and a set of precise body measurements. The body measurements included height, reach, width of head, and the length of the left foot.
The Computation of Limits in Elementary Calculus
Authors: Stenger William
Published: January 2021

Beginning calculus students often experience difficulty in computing limits at a point where the function is undefined or where the function has different defining formulas on the left and on the right.
An Art-Full Application Using Magic Squares
Authors: Kenney Margaret J.
Published: January 2021

Blending mathematics and art can be an exciting classroom experience for all students. Essentially there are two types of activities for the mathematics teacher to consider using. On the one hand, units can be developed that uncover and emphasize the mathematics that is present in existing works of art. A viable alternative, which promotes student creativity, is the actual development of designs based on some specific mathematical pattern. This article presents a development of student-made designs using one particular number pattern-the mbgic square. Students, provided with a supply of magic squares and some simply stated rules for producing design elements, will be able to create some striking art.
A Unit on Slope Functions-Using a Computer in Mathematics Class
Authors: Lappan Glenda , Winter M. J.
Published: January 2021

The slope of a function is a fundamental concept in mathematics. This article de- scribes an introductory unit on slope designed to give students a chance to discover some of the basic relationships between functions and slopes. The unit is appropriate for any senior level course for collegebound students. We use the graphics on a microcomputer to provide students with a tool to generate many examples. Students use the program “Slope and Graph” and their own programs to investigate properties and applications of slope functions. ("Slope and Graph" was written for an 8K PET. A copy may be obtained by sending a blank cassette and self-addressed, stamped envelope to us.)
Clubs: A Student-Run Geometry Contest
Authors: Ames Charles G.
Published: January 2021

In May I970, the Newfane Senior High School Math Club (Newfane, New York) instituted a Plane Geometry Contest which has become an annual affair. The contest, now in its twelfth year, is held in the spring. The examination consists of seventy-one multiple-choice questions and four construction problems. Only for-ty-five minutes are allotted for the test. Speed and accuracy are stressed. The contest is scored by awarding one point for a correct response and deducting one-quarter point for an incorrect response. Unanswered questions are not scored. Although there are four responses listed for each question, occasionally the correct response is not listed, and some questions have incomplete or inconsistent information. The responses are labeled A, B, C, D, with E used for the situations just mentioned. Questions have a wide range of difficulty, with some being straightforward, routine problems, whereas others require a great deal of thought and manipulation.
Clubs: Computer Clubs = Computer Education Programs
Authors: Spencer Marilyn J.
Published: January 2021

Join The Movement (see References) was an exciting new program sponsored by The Association of Computing Machinery (ACM) to establish and support a Secondary Computer Club Program through ACM chapter memberships. In 1971, under the sponsorship of the Northern New Jersey ACM chapter, George G. Heller, who was the founder of the “movement,” conducted a successful kickoff campaign at a host New Jersey school. This program was attended by nearly four hundred people, which included students, administrators, teachers, parents, and ACM members. Following this presentation, fifteen schools in Northern New Jersey joined the movement by forming computer clubs, with each club having a computer-professional ACM advisor in addition to their local school advisor.
Applications: Prediction Runs
Authors: Zitarelli David E.
Published: January 2021

My local jogging club, the Delco RRC, holds weekly fun runs around a three-mile loop. In order to add some spice to our weekly routine, we sometimes hold “prediction runs,” in which the winner is the runner who comes closest to predicting the total time it will take him or her to complete the loop. Watches are not allowed. A variation on this theme is the “couple prediction run," where two people, usually one male and one female, run in opposite directions until they meet, at which point they touch hands, reverse their directions, and run back to the starting point. The couple that finishes closest to its predicted time wins.
Emmy Noether, Greatest Woman Mathematician
Authors: Kimberling Clark
Published: January 2021

Emmy Noether was born one hundred years ago in the German university town of Erlangen, where her father. Max Noether, was a professor of mathematics. At that time it was very unusual for a woman to seek a university education. In fact, a leading historian of the day wrote that talk of “surrendering our universities to the invasion of women… is a shameful display of moral weakness.”1 At the University of Erlangen, the Academic Senate in 1898 declared that the admission of women students would “overthrow all academic order.”2
Regular Polygons and Geometric Series
Authors: Jarrett Joscelyn A.
Published: January 2021

The limit concept is among the more important topics in mathematics. Yet, because it is often taught very abstractly, students do not have a good intuitive feeling for the concept. They can benefit from more concrete illustrations. Geometric applications serve as a vehicle for making limit concepts more meaningful. Recent articles by Kich (1979) and Van Beynen and McGinty (1981) provide some interesting geometric illustrations of limits. The following examples are some of my favorites.
A New Angle For Constructing Pentagons
Authors: Benson John , Borkovitz Debra
Published: January 2021

The traditional high school geometry class can be enhanced by the addition of appropriate problem-solving activities. One such problem, the construction of a pentagon, can be divided into three worth-while tasks.
Constructions With An Unmarked Protractor
Authors: Austin Joe Dan
Published: January 2021

Instead of reviewing familiar constructions, give your able students some challenging new ones.
Giving Geometry Students An Added Edge In Constructions
Authors: Gibb Allan A.
Published: January 2021

Several recent articles in this journal and in other NCTM publications indicate a continuing interest in alternatives to the straightedge-and-compass constructions of traditional high school geometry (see Bibliography). A description of various construction tools and techniques may be found in the literature (Eves 1963; Yates 1949), and the history of the topic from the time of the Greeks to the present is a fascinating one. Of the simple construction instruments, one of the least well known is the ruler with two parallel straightedges (a parallel ruler). In 1890, Adler (1906) showed that all the ruler-and-compass constructions can be done using only a ruler with two edges at a fixed angle; the parallel ruler is the special case with angle zero. There does not seem to be a treatment of constructions with the parallel ruler that is at once reasonably complete and accessible to high school geometry students. Most discussions make use of terminology and theorems beyond that usually covered at the high school level, and, further, much of even this material, such as Yates’s Geometrical Tools, is now out of print—surely an undeserved fate for so delightful a book.
Circles, Polar Graphs, And A Computer—Some Unexpected Results
Authors: Wagner William J.
Published: January 2021

What short BASIC program written by a high school freshman can produce all the displays shown in figure 1 and more? This article discusses a remarkably creative effort by two students, ninth grader Ryoji Watanabe and tenth grader Jerry Hogsett, who went as far as they did with their ideas because of their willingness to experiment. because of the availability of flexible computer graphics, and because they had not been taught the traditional ways of trigonometry.